analytical geometry: 2d and 3d
TRANSCRIPT
ANALYTICALGEOMETRY
2Dand3D
P.R.VittalVisitingProfessor
DepartmentofStatisticsUniversityofMadras
Chennai
Chennai•Delhi
BriefContents
AbouttheAuthor
Preface
1CoordinateGeometry
2TheStraightLine
3PairofStraightLines
4Circle
5SystemofCircles
6Parabola
7Ellipse
8Hyperbola
9PolarCoordinates
10TracingofCurves
11ThreeDimension
12Plane
13StraightLine
14Sphere
15Cone
16Cylinder
Contents
AbouttheAuthor
Preface
1CoordinateGeometry
1.1Introduction
1.2SectionFormula
IllustrativeExamples
Exercises
2TheStraightLine
2.1Introduction
2.2SlopeofaStraightLine
2.3Slope-interceptFormofaStraightLine
2.4InterceptForm
2.5Slope-pointForm
2.6TwoPointsForm
2.7NormalForm
2.8ParametricFormandDistanceForm
2.9PerpendicularDistanceonaStraightLine
2.10IntersectionofTwoStraightLines
2.11ConcurrentStraightLines
2.12AnglebetweenTwoStraightLines
2.13EquationsofBisectorsoftheAnglebetweenTwoLines
IllustrativeExamples
Exercises
3PairofStraightLines
3.1Introduction
3.2HomogeneousEquationofSecondDegreeinxandy
3.3AnglebetweentheLinesRepresentedbyax2+2hxy+by2=0
3.4EquationfortheBisectoroftheAnglesbetweentheLinesGivenbyax2+2hxy+by2=0
3.5ConditionforGeneralEquationofaSecondDegreeEquationtoRepresentaPairofStraightLines
IllustrativeExamples
Exercises
4Circle
4.1Introduction
4.2EquationofaCirclewhoseCentreis(h,k)andRadiusr
4.3CentreandRadiusofaCircleRepresentedbytheEquationx2+y2+2gx+2fy+c=0
4.4LengthofTangentfromPointP(x1,y1)totheCirclex2+y2+2gx+2fy+c=0
4.5EquationofTangentat(x1,y1)totheCirclex2+y2+2gx+2fy+c=0
4.6EquationofCirclewiththeLineJoiningPointsA(x1,y1)andB(x2,y2)astheendsofDiameter
4.7ConditionfortheStraightLiney=mx+ctobeaTangenttotheCirclex2+y2=a2
4.8EquationoftheChordofContactofTangentsfrom(x1,y1)totheCirclex2+y2+2gx+2fy+c
=0
4.9TwoTangentscanAlwaysbeDrawnfromaGivenPointtoaCircleandtheLocusofthePointofIntersectionofPerpendicularTangentsisaCircle
4.10PoleandPolar
4.11ConjugateLines
4.12EquationofaChordofCirclex2+y2+2gx+2fy+c=0inTermsofitsMiddlePoint
4.13CombinedEquationofaPairofTangentsfrom(x1,y1)totheCirclex2+y2+2gx+2fy+c=
0
4.14ParametricFormofaCircle
IllustrativeExamples
Exercises
5SystemofCircles
5.1RadicalAxisofTwoCircles
5.2OrthogonalCircles
5.3CoaxalSystem
5.4LimitingPoints
5.5Examples(RadicalAxis)
5.6Examples(LimitingPoints)
Exercises
6Parabola
6.1Introduction
6.2GeneralEquationofaConic
6.3EquationofaParabola
6.4LengthofLatusRectum
6.5DifferentFormsofParabola
IllustrativeExamplesBasedonFocusDirectrixProperty
6.6ConditionforTangency
6.7NumberofTangents
6.8PerpendicularTangents
6.9EquationofTangent
6.10EquationofNormal
6.11EquationofChordofContact
6.12PolarofaPoint
6.13ConjugateLines
6.14PairofTangents
6.15ChordIntermsofMid-point
6.16ParametricRepresentation
6.17ChordJoiningTwoPoints
6.18EquationsofTangentandNormal
6.19PointofIntersectionofTangents
6.20PointofIntersectionofNormals
6.21NumberofNormalsfromaPoint
6.22IntersectionofaParabolaandaCircle
IllustrativeExamplesBasedonTangentsandNormals
IllustrativeExamplesBasedonParameters
Exercises
7Ellipse
7.1StandardEquation
7.2StandardEquationofanEllipse
7.3FocalDistance
7.4PositionofaPoint
7.5AuxiliaryCircle
IllustrativeExamplesBasedonFocus-directrixProperty
7.6ConditionforTangency
7.7DirectorCircleofanEllipse
7.8EquationoftheTangent
7.9EquationofTangentandNormal
7.10EquationtotheChordofContact
7.11EquationofthePolar
7.12ConditionforConjugateLines
IllustrativeExamplesBasedonTangents,Normals,Pole-polarandChord
7.13EccentricAngle
7.14EquationoftheChordJoiningthePoints
7.15EquationofTangentat‘θ’ontheEllipse
7.16ConormalPoints
7.17ConcyclicPoints
7.18EquationofaChordinTermsofitsMiddlePoint
7.19CombinedEquationofPairofTangents
7.20ConjugateDiameters
7.21Equi-conjugateDiameters
IllustrativeExamplesBasedonConjugateDiameters
Exercises
8Hyperbola
8.1Definition
8.2StandardEquation
8.3ImportantPropertyofHyperbola
8.4EquationofHyperbolainParametricForm
8.5RectangularHyperbola
8.6ConjugateHyperbola
8.7Asymptotes
8.8ConjugateDiameters
8.9RectangularHyperbola
Exercises
9PolarCoordinates
9.1Introduction
9.2DefinitionofPolarCoordinates
9.3RelationbetweenCartesianCoordinatesandPolarCoordinates
9.4PolarEquationofaStraightLine
9.5PolarEquationofaStraightLineinNormalForm
9.6Circle
9.7PolarEquationofaConic
Exercises
10TracingofCurves
10.1GeneralEquationoftheSecondDegreeandTracingofaConic
10.2ShiftofOriginwithoutChangingtheDirectionofAxes
10.3RotationofAxeswithoutChangingtheOrigin
10.4RemovalofXY-term
10.5Invariants
10.6ConditionsfortheGeneralEquationoftheSecondDegreetoRepresentaConic
10.7CentreoftheConicGivenbytheGeneralEquationoftheSecondDegree
10.8EquationoftheConicReferredtotheCentreasOrigin
10.9LengthandPositionoftheAxesoftheCentralConicwhoseEquationisax2+2hxy+by2=1
10.10AxisandVertexoftheParabolawhoseEquationisax2+2hxy+by2+2gx+2fy+c=0
Exercises
11ThreeDimension
11.1RectangularCoordinateAxes
11.2FormulaforDistancebetweenTwoPoints
11.3CentroidofTriangle
11.4CentroidofTetrahedron
11.5DirectionCosines
IllustrativeExamples
Exercises
12Plane
12.1Introduction
12.2GeneralEquationofaPlane
12.3GeneralEquationofaPlanePassingThroughaGivenPoint
12.4EquationofaPlaneinInterceptForm
12.5EquationofaPlaneinNormalForm
12.6AnglebetweenTwoPlanes
12.7PerpendicularDistancefromaPointonaPlane
12.8PlanePassingThroughThreeGivenPoints
12.9ToFindtheRatioinwhichthePlaneJoiningthePoints(x1,y1,z1)and(x2,y2,z2)isDividedbythePlaneax+by+cz+d=0.
12.10PlanePassingThroughtheIntersectionofTwoGivenPlanes
12.11EquationofthePlaneswhichBisecttheAnglebetweenTwoGivenPlanes
12.12ConditionfortheHomogenousEquationoftheSecondDegreetoRepresentaPairofPlanes
IllustrativeExamples
Exercises
13StraightLine
13.1Introduction
13.2EquationofaStraightLineinSymmetricalForm
13.3EquationsofaStraightLinePassingThroughtheTwoGivenPoints
13.4EquationsofaStraightLineDeterminedbyaPairofPlanesinSymmetricalForm
13.5AnglebetweenaPlaneandaLine
13.6ConditionforaLinetobeParalleltoaPlane
13.7ConditionsforaLinetoLieonaPlane
13.8ToFindtheLengthofthePerpendicularfromaGivenPointonaLine
13.9CoplanarLines
13.10SkewLines
13.11EquationsofTwoNon-intersectingLines
13.12IntersectionofThreePlanes
13.13ConditionsforThreeGivenPlanestoFormaTriangularPrism
IllustrativeExamples
IllustrativeExamples(CoplanarLinesandShortestDistance)
Exercises
14Sphere
14.1DefinitionofSphere
14.2Theequationofaspherewithcentreat(a,b,c)andradiusr
14.3EquationoftheSphereontheLineJoiningthePoints(x1,y1,z1)and(x2,y2,z2)asDiameter
14.4LengthoftheTangentfromP(x1,y1,z1)totheSpherex2+y2+z2+2ux+2vy+2wz+d=0
14.5EquationoftheTangentPlaneat(x1,y1,z1)totheSpherex2+y2+z2+2ux+2vy+2wz+d
=0
14.6SectionofaSpherebyaPlane
14.7EquationofaCircle
14.8IntersectionofTwoSpheres
14.9EquationofaSpherePassingThroughaGivenCircle
14.10ConditionforOrthogonalityofTwoSpheres
14.11RadicalPlane
14.12CoaxalSystem
IllustrativeExamples
Exercises
15Cone
15.1DefinitionofCone
15.2EquationofaConewithaGivenVertexandaGivenGuidingCurve
15.3EquationofaConewithitsVertexattheOrigin
15.4ConditionfortheGeneralEquationoftheSecondDegreetoRepresentaCone
15.5RightCircularCone
15.6TangentPlane
15.7ReciprocalCone
Exercises
16Cylinder
16.1Definition
16.2EquationofaCylinderwithaGivenGeneratorandaGivenGuidingCurve
16.3EnvelopingCylinder
16.4RightCircularCylinder
IllustrativeExamples
Exercises
AbouttheAuthor
P.R.VittalwasapostgraduateprofessorofMathematicsatRamakrishnaMissionVivekanandaCollege,Chennai,fromwhereheretiredasPrincipalin1996.HewasavisitingprofessoratWesternCarolinaUniversity,USA,andhasvisitedanumberofuniversitiesintheUSAandCanadainconnectionwithhisresearchwork.Heis,atpresent,avisitingprofessorattheDepartmentofStatistics,UniversityofMadras;InstituteofCharteredAccountantsofIndia,Chennai;TheInstituteofTechnologyandManagement,Chennai;andNationalManagementSchool,Chennai,besidesbeingaresearchguideinManagementScienceatBITS,Ranchi.ProfessorVittalhaspublished30researchpapersinjournalsofnationaland
internationalreputeandguidedanumberofstudentstotheirM.Phil.andPh.D.degrees.AfellowofTamilNaduAcademyofSciences,hisresearchtopicsareprobability,stochasticprocesses,operationsresearch,differentialequationsandsupplychainmanagement.Hehasauthoredabout30booksinmathematics,statisticsandoperationsresearch.
TomygrandchildrenAaravandAdvay
Preface
Asuccessfulcourseinanalyticalgeometrymustprovideafoundationforfutureworkinmathematics.Ourteachingresponsibilitiesaretoinstilcertaintechnicalcompetenceinourstudentsinthisdisciplineofmathematics.Agoodtextbook,aswithagoodteacher,shouldaccomplishtheseaims.Inthisbook,youwillfindacrisp,mathematicallyprecisepresentationthatwillallowyoutoeasilyunderstandandgraspthecontents.Thisbookcontainsbothtwo-dimensionalandthree-dimensionalanalytical
geometry.Insomeofthefundamentalresults,vectortreatmentisalsogivenandtherefromthescalarformoftheresultshasbeendeduced.Thefirst10chaptersdealwithtwo-dimensionalanalyticalgeometry.In
Chapter1,allbasicresultsareintroduced.Theconceptoflocusiswellexplained.Usingthisidea,inChapter2,differentformsfortheequationofastraightlineareobtained;allthecharacteristicsofastraightlinearealsodiscussed.Chapter3dealswiththeequationofapairofstraightlinesanditsproperties.InChapters4and5,circleandsystemofcircles,includingcoaxialsystemandlimitingpointsofacoaxialsystem,areanalysed.Chapters6,7and8dealwiththeconicsections—parabola,ellipseand
hyperbola.Apartfromtheirpropertiessuchasfocusanddirectrix,theirparametricequationsarealsoexplained.Specialpropertiessuchasconormalpointsofallconicsaredescribedindetails.Conjugatediametersinellipseandhyperbolaandasymptotesofahyperbolaandrectangularhyperbolaarealsoanalysedwithanumberofexamples.Ageneraltreatmentofconicsandtracingofconicsisalsoprovided.InChapter9,wedescribepolarcoordinates,whichareusedtomeasure
distancesforsomespecialpurposes.Chapter10examinestheconditionsforthegeneralequationoftheseconddegreetorepresentthedifferenttypesofconics.
InChapters11to16,westudythethree-dimensionalanalyticalgeometry.Thebasicconcepts,suchasdirectionalcosines,areintroducedinChapter11.InChapter12,allformsofplaneareanalysedwiththehelpofexamples.Chapter13introducesastraightlineasanintersectionofapairofplanes.Differentformsofastraightlinearestudied;especially,coplanarlinesandtheshortestdistancebetweentwoskewlines.Chapter14dealswithspheresandsystemofspheres.InChapters15and16,twospecialtypesofconicoids—coneandcylinder—arediscussed.Anumberofillustrativeexamplesandexercisesforpracticearegiveninall
these16chapters,tohelpthestudentsunderstandtheconceptsinabettermanner.Ihopethatthisbookwillbeveryusefulforundergraduatestudentsand
engineeringstudentswhoneedtostudyanalyticalgeometryaspartoftheircurriculum.
Chapter1
CoordinateGeometry
1.1INTRODUCTION
LetXOX′andYOY′betwofixedperpendicularlinesintheplaneofthepaper.ThelineOXiscalledtheaxisofXandOYtheaxisofY.OXandOYtogetherarecalledthecoordinateaxes.ThepointOiscalledtheoriginofthecoordinateaxes.LetPbeapointinthisplane.DrawPMperpendiculartoXOX′.ThedistanceOMiscalledthex-coordinateorabscissaandthedistanceMPiscalledthey-coordinateorordinateofthepointP.
IfOM=xandMP=ythen(x,y)arecalledthecoordinatesofthepointP.ThecoordinatesoftheoriginOare(0,0).ThelinesXOX′andYOY′dividetheplaneintofourquadrants.TheyareXOY,YOX′,X′OY′andY′OX′.ThelengthsmeasuredinthedirectionsOXandOYareconsideredpositiveandthelengthsmeasuredinthedirectionsOX′andOY′areconsiderednegative.Thenatureofthecoordinatesinthedifferentquadrantsisasfollows:
Quadrant x-coordinate y-coordinate
First+ +
Second− +
− +
Third− −
Fourth+ −
ThemethodofrepresentingapointbymeansofcoordinateswasfirstintroducedbyRenaDescartesandhencethisbranchofmathematicsiscalledtherectangularCartesiancoordinatesystem.Usingthiscoordinatesystem,onecaneasilyfindthedistancebetweentwo
pointsinaplane,thecoordinatesofthepointthatdividesalinesegmentinagivenratio,thecentroidofatriangle,theareaofatriangleandthelocusofapointthatmovesaccordingtoagivengeometricallaw.
1.1.1DistancebetweenTwoGivenPoints
LetPandQbetwopointswithcoordinates(x1,y1)and(x2,y2).
DrawPLandQMperpendicularstothex-axis,anddrawQNperpendiculartoPL.Then,
Note1.1.1:ThedistanceofPfromtheoriginOis
Example1.1.1
IfPisthepoint(4,7)andQis(2,3),then
Example1.1.2
ThedistancebetweenthepointsP(2,−5)andQ(−4,7)is
1.2SECTIONFORMULA
1.2.1CoordinatesofthePointthatDividestheLineJoiningTwoGivenPointsinaGivenRatio
LetthetwogivenpointsbeP(x1,y1)andQ(x2,y2).
LetthepointRdividePQinternallyintheratiol:m.DrawPL,QMandRNperpendicularstothex-axis.DrawPSperpendiculartoRNandRTperpendiculartoMQ.LetthecoordinatesofRbe(x,y).RdividesPQinternallyintheratiol:m.Then,
TrianglesPSRandRTQaresimilar.
Also
Hence,thecoordinatesofRare
1.2.2ExternalPointofDivision
IfthepointR′dividesPQexternallyintheratiol:m,then
Choosingmnegative,wegetthecoordinatesofR′.Therefore,thecoordinatesof
R′are
Note1.2.2.1:Ifwetakel=m=1intheinternalpointofdivision,wegetthecoordinatesofthemidpoint.Therefore,thecoordinatesofthemidpointofPQ
are
1.2.3CentroidofaTriangleGivenitsVertices
LetABCbeatrianglewithverticesA(x1,y1),B(x2,y2)andC(x3,y3).
LetAA′,BB′andCC′bethemediansofthetriangle.ThenA′,B′,C′arethemidpointsofthesidesBC,CAandAB,respectively.ThecoordinatesofA′are
Weknowthatthemediansofatriangleareconcurrentatthe
pointGcalledthecentroidandGdivideseachmedianintheratio2:1.ConsideringthemedianAA′,thecoordinatesofGare
1.2.4AreaofTriangleABCwithVerticesA(x1,y1),B(x2,y2)andC(x3,y3)
LettheverticesoftriangleABCbeA(x1,y1),B(x2,y2)andC(x3,y3).
DrawAL,BMandCNperpendicularstoOX.Then,areaΔoftriangleABCiscalculatedas
Note1.2.4.1:Theareaispositiveornegativedependingupontheorderinwhichwetakethepoints.Sincescalarareaisalwaystakentobeapositivequantity,wetake
Note1.2.4.2:Iftheverticesofthetriangleare(0,0),(x1,y1)and(x2,y2),then
Note1.2.4.3:Iftheareaofthetriangleiszero,i.e.Δ=0,thenwenotethatthepointsarecollinear.Hence,theconditionforthepoints(x1,y1),(x2,y2)and(x3,y3)tobecollinearis
x1(y2−y3)+x2(y3−y1)+x3(y1−y2)=0
1.2.5AreaoftheQuadrilateralGivenitsVertices
LetABCDbethequadrilateralwithverticesA(x1,y1),B(x2,y2),C(x3,y3)andD(x4,y4).DrawAP,BQ,CRandDSperpendicularstothex-axis.Then,
Note1.2.5.1:Thisresultcanbeextendedtoapolygonofnsideswithvertices(x1,y1),(x2,y2)……(xn,yn)as
Locus
Whenapointmovessoastosatisfysomegeometricalconditionorconditions,thepathtracedoutbythepointiscalledthelocusofthepoint.
thepathtracedoutbythepointiscalledthelocusofthepoint.Forexample,ifapointmoveskeepingaconstantdistancefromafixedpoint,
thelocusofthemovingpointiscalledcircleandthefixeddistanceiscalledtheradiusofthecircle.Moreover,ifapointmovessuchthatitsdistancefromtwofixedpointsareequal,thenthelocusofthepointistheperpendicularbisectorofthelinejoiningthetwofixedpoints.IfAandBaretwofixedpointsandpointP
movessuchthat thenthelocusofPisacirclewithABasthe
diameter.Itispossibletorepresentthelocusofapointbymeansofanequation.SupposeapointP(x,y)movessuchthatitsdistancefromtwofixedpoints
A(2,3)andB(5,−3)areequal.ThenthegeometricallawisPA=PB⇒PA2=PB2
Here,thelocusofPisastraightline.
ILLUSTRATIVEEXAMPLES
Example1.1
Findthedistancebetweenthepoints(4,7)and(−2,5).
Solution
LetPandQbethepoints(4,7)and(−2,5),respectively.
Example1.2
Provethatthepoints(4,3),(7,−1)and(9,3)aretheverticesofanisoscelestriangle.
Solution
LetA(4,3),B(7,−1),C(9,3)bethethreegivenpoints.Then
Sincethesumoftwosidesisgreaterthanthethird,thepointsformatriangle.Moreover,AB=AC=5.Therefore,thetriangleisanisoscelestriangle.
Example1.3
Showthatthepoints(6,6),(2,3)and(4,7)aretheverticesofarightangledtriangle.
Solution
LetA,B,Cbethepoints(6,6),(2,3)and(4,7),respectively.
Hence,thepointsaretheverticesofarightangledtriangle.
Example1.4
Showthatthepoints(7,9),(3,−7)and(−3,3)aretheverticesofarightangledisoscelestriangle.
Solution
LetA,B,Cbethepoints(7,9),(3,−7),(−3,3),respectively.
Hence,thepointsareverticesofarightangledtriangle.Also,BC=AC.Therefore,itisarightangledisoscelestriangle.
Example1.5
Showthatthepoints(4,−4),(−4,4)and aretheverticesofan
equilateraltriangle.
Solution
LetA,B,Cbethepoints(4,−4),(−4,4)and ,respectively.
Hence,thepointsA,BandCaretheverticesofanequilateraltriangle.
Example1.6
Showthatthesetofpoints(−2,−1),(1,0),(4,3)and(1,2)aretheverticesofaparallelogram.
Solution
LetA,B,C,Dbethepoints(−2,−1),(1,0),(4,3)and(1,2),respectively.Aquadrilateralisaparallelogramiftheoppositesidesareequal.
SincetheoppositesidesofthequadrilateralABCDareequal,thefourpointsformaparallelogram.
Example1.7
Showthatthepoints(2,−2),(8,4),(5,7)and(−1,1)aretheverticesofarectangletakeninorder.
Solution
Aquadrilateralinwhichtheoppositesidesareequalandthediagonalsareequalisarectangle.LetA(2,−2),B(8,4),C(5,7)andD(−1,1)bethefourgivenpoints.
Thus,theoppositesidesareequalandthediagonalsarealsoequal.Hence,thefourpointsformarectangle.
Example1.8
Provethatthepoints(3,2),(5,4),(3,6)and(1,4)takeninorderformasquare.
Solution
Aquadrilateralinwhichallsidesareequalanddiagonalsareequalisasquare.LetA,B,C,Dbethepoints(3,2),(5,4),(3,6),(1,4),respectively.
Thus,allsidesareequalandalsothediagonalsareequal.Hence,thefourpointsformasquare.
Example1.9
FindthecoordinatesofthecircumcentreofatrianglewhoseverticesareA(3,−2),B(4,3)andC(−6,5).Also,findthecircumradius.
Solution
LetA(3,−2),B(4,3),andC(−6,5)bethegivenpoints.LetS(x,y)bethecircumcentreofΔABC.ThenSA=SB=SC=circumradius.Now
Hence,thecircumcentreis
Now
Therefore,circumradius units.
Example1.10
Showthatthepoints(3,7),(6,5)and(15,−1)lieonastraightline.
Solution
LetA(3,7),B(6,5)andC(15,−1)bethethreepoints.Then
Hence,thethreegivenpointslieonastraightline.
Example1.11
Showthat(4,3)isthecentreofthecirclethatpassesthroughthepoints(9,3),(7,−1)and(1,−1).Finditsradius.
Solution
LetA(9,3),B(7,−1),C(1,−1)andP(4,3)bethegivenpoints.Then
Hence,PisthecentreofthecirclepassingthroughthepointsA,B,C;itsradiusis5.
Example1.12
IfOistheoriginandthecoordinatesofAandBare(x1,y1)and(x2,y2),
respectively,provethatOA·OBcosθ=x1x2+y1y2where
Solution
Bycosineformula
Hence,OA·OBcosθ=x1x2+y1y2
Example1.13
Iftanα,tanβandtanγbetherootsoftheequationx3−3ax2+3bx−c=0andtheverticesofthetriangleABCare(tanα,cotα),(tanβ,cotβ)and(tanγ,cotγ)showthatthecentroidofthetriangleis(a,b).
Solution
Giventaα,tanβandtanγaretherootsoftheequation
Thendividing(1.5)by(1.6),
ThecentroidofΔABCis
(i.e.)(a,b)from(1.4)and(1.7).
Example1.14
Iftheverticesofatrianglehaveintegralcoordinates,provethatitcannotbeanequilateraltriangle.
Solution
Theareaofthetrianglewithvertices(x1,y1),(x2,y2)and(x3,y3)is
Also,theareaofΔABCis
whereaisthesideoftheequilateraltriangle.Iftheverticesofthetrianglehaveintegralcoordinates,thenΔisarationalnumber.However,from(1.9)weinfer
thattheareais timesarationalnumber.Hence,iftheverticesofatriangle
haveintegralcoordinates,itcannotbeequilateral.
Example1.15
Ift1,t2andt3aredistinct,thenshowthatthepoints and
a≠0cannotbecollinear.
Solution
sincet1,t2andt3aredistinct.Hence,thethreegivenpointscannotbecollinear.
Example1.16
TheverticesofatriangleABCare(2,3),(4,7),(−5,2).FindthelengthofthealtitudethroughA.
Solution
TheareaofΔABCisgivenby
Weknowthat
Example1.17
TheverticesofatriangleABCareA(x1,x1tanα),B(x2,x2tanβ)andC(x3,x3
tanγ).If istheorthocentreandS(0,0)isthecircumcentre,thenshowthat
Solution
IfristhecircumradiusofΔABC,SA=SB=SC=r,SA2=r2
Then,thecoordinatesofA,BandCare(rcosα,rsinα),(rcosβ,rsinβ)and(rcosγ,rsinγ).Thecentroidofthetriangleis
Theorthocentreis andthecircumcentreisS(0,0).Geometricallywe
knowthatH,GandSarecollinear.Therefore,theslopeofSGandGHareequal.
Example1.18
AlinejoiningthetwopointsA(2,0)andB(3,1)isrotatedaboutAintheanticlockwisedirectionthroughanangleof15°.IfBgoestoCinthenewposition,findthecoordinateofC.
Solution
GiventhatCisthenewpositionofB.DrawCLperpendiculartoOXandlet(x1,y1)bethecoordinatesofC.Now
ABmakes45°withx-axisand Then
Hence,thecoordinatesofCare
Example1.19
ThecoordinatesofA,BandCare(6,3),(−3,5)and(4,−2),respectively,andP
isanypoint(x,y).ShowthattheratiooftheareaofΔPBCandΔABCis
Solution
ThepointsA,B,CandPare(6,3),(−3,5),(4,−2)and(x,y),respectively.
Example1.20
Findthecoordinatesofthepointthatdividesthelinejoiningthepoints(2,3)and(−4,7)(i)internally(ii)externallyintheratio3:2.
Solution
LetRandR′respectivelydividePQinternallyandexternallyintheratio3:2.
i. ThecoordinatesofRare
ii. ThecoordinatesofR′are
(i.e)(−16,15)
Example1.21
Findtheratioinwhichthelinejoiningthepoints(4,7)and(−3,2)isdividedbythey-axis.
Solution
Letthey-axismeetthelinejoiningthejointsP(4,7)andQ(−3,2)atR.LetthecoordinatesofRbe(0,y).LetRdividePQintheratiok:1.
ThecoordinatesofRisgivenby
Hence,theratioinwhichRdividesPQis4:3.
Example1.22
Showthatthepoints(−2,−1),(1,0),(4,3)and(1,2)formtheverticesofaparallelogram.
Solution
Aquadrilateralinwhichthediagonalsbisecteachotherisaparallelogram.
ThemidpointofACis (i.e.)(1,1).
ThemidpointofBDis (i.e.)(1,1).
Sincethediagonalsbisecteachother,ABCDisaparallelogram.
Example1.23
Find(x,y)if(3,2),(6,3),(x,y)and(6,5)aretheverticesofaparallelogramtakeninorder.
Solution
LetthefourpointsbeA,B,CandD,respectively.SinceABCDisaparallelogram,themidpointofACisthesameasthemidpointofBD.
ThemidpointofACis .ThemidpointofBDis (i.e)(6,4).
Hence(x,y)is(9,6).
Example1.24
Themidpointsofthesidesofatriangleare(6,−1),(−1,−2)and(1,4).Findthecoordinatesofthevertices.
Solution
LetD,EandFbethemidpointsofthesidesBC,CAandAB,respectively.Then,(6,−1),(−1,−2),(1,4)arethepointsD,EandF,respectively.LetA(x1,y1),B(x2,y2)andC(x3,y3)betheverticesofthetriangle.ThenBDEFisaparallelogram.
ThemidpointofDFis
ThemidpointofBEis
SinceFisthemidpointofAB,
SinceEisthemidpointofAC,
∴x3=4,y3=−7.∴Cisthepoint(4,−7)
Hence,theverticesofthetriangleare(−6,3),(8,5)and(4,−7).
Example1.25
Showthattheaxesofcoordinatestrisectthestraightlinejoiningthepoints(2,−2)and(−1,4).
Solution
Letthelinejoiningthepoints(2,−2)and(−1,4)meetx-axisandy-axisatAandB,respectively.LetthecoordinatesofAandBbe(x,0)and(0,y),respectively.LetAdividethelineintheratiok:1.Thenthex-coordinateofAisgivenby
∴−k+2=0⇒k=2Hence,Adividesthelineintheratio2:1.
LetBdividethelineintheratiol:1.Then,
Hence,
Bdividesthelineintheratio1:2.Hence,AandBtrisectthelinejoiningthepoints(2,−4)and(−1,4).
Example1.26
TheverticesofatriangleareA(3,5),B(−7,9)andC(1,−3).Findthelengthofthethreemediansofthetriangle.
Solution
LetD,EandFbethemidpointsofthesidesofBC,CAandAB,respectively.
ThecoordinatesofDare (i.e.)(−3,3).ThecoordinatesofEare
(i.e.)(2,1).ThecoordinatesofFare (i.e.)(−2,7).
Hence,thelengthsofthemediansare
units.
Example1.27
Twooftheverticesofatriangleare(4,7)and(−1,2)andthecentroidisattheorigin.Findthethirdvertex.
Solution
Letthethirdvertexofthetrianglebe(x,y).Then
Hence,thethirdvertexis(−3,−9).
Example1.28
Showthatthemidpointofthehypotenuseoftherightangledtrianglewhoseverticesare(8,−10),(7,−3)and(0,−4)isequidistantfromthevertices.
Solution
LetthethreegivenpointsbeA(8,−10),B(7,−3)andC(0,−4).
AB2+BC2=AC2
Hence,ABCisarightangledtrianglewithACashypotenuse.
ThemidpointofACis (i.e)(4,−7).
Hence,themidpointofthehypotenuseisequidistantfromthevertices.
Example1.29
Findtheratioinwhichthelinejoiningthepoints(1,−1)and(4,5)isdividedbythepoint(2,1).
Solution
LetthepointR(2,1)dividethelinejoiningthepointsP(1,−1)andQ(4,5)intheratiok:1.Then
Therefore,R(2,1)dividesPQintheratio1:2.
Example1.30
Findthelocusofthepointthatisequidistantfromtwogivenpoints(2,3)and(−4,1).
Solution
LetP(x,y)beapointsuchthatPA=PBwhereAandBarethepoints(2,3)and(−4,1),respectively.
Example1.31
Findthelocusofthepointthatmovesfromthepoint(4,3)keepingaconstantdistanceof5unitsfromit.
Solution
LetC(4,3)bethegivenpointandP(x,y)beanypointsuchthatCP=5.Then
Example1.32
Theendsofarodoflengthlmoveontwomutuallyperpendicularlines.Showthatthelocusofthepointontherodthatdividesitintheratio1:2is9x2+36y2=l2.
Solution
LetABbearodoflengthlwhoseendsAandBareonthecoordinateaxes.LetthecoordinatesofAandBbeA(a,0)andB(0,b).LetthepointP(x1,y1)divideABintheratio1:2.
ThenthecoordinatesofPare
Hence,thelocusof(x1,y1)is9x2+36y2=l2.
Example1.33
Apointmovessuchthatthesumofitsdistancesfromtwofixedpoints(al,0)
and(−al,0)isalways2a.Provethattheequationofthelocusis
Solution
LetthetwofixedpointsbeA(al,0)andB(0,−al).LetP(x1,y1)beamovingpointsuchthatPA+PB=2a.Giventhat
Then
Adding(1.10)and(1.11),
Squaringonbothsides,weget
Dividingbya2(1−l2),weget
Therefore,thelocusof(x1,y1)is
Example1.34
ArightangledtrianglehavingtherightangleatCwithCA=aandCB=bmovessuchthattheangularpointsAandBslidealongthex-axisandy-axis,respectively.FindthelocusofC.
Solution
LetthepointsAandBbeonthex-axisandy-axis,respectively.LetAandBhavecoordinates(α,0)and(0,β).LetCbethepointwithcoordinates(x1,y1).
Then
ThenAB2=a2+b2.AlsoAB2=α2+β2
Hence,α2+β2=a2+b2
Hence,thelocusofc(x1,y1)isa2x2−b2y2=0.
Example1.35
TwopointsPandQaregiven.RisavariablepointononesideofthelinePQ
suchthat isapositiveconstant2α.FindthelocusofthepointP.
Solution
LetPQbethex-axisandtheperpendicularthroughthemidpointofPQbethey-axis.LetPandQbethepoints(a,0)and(−a,0),respectively.LetRbethepoint
(x1,y1).Let Then
(i.e.)θ−ϕ=2α.Thentan(θ−ϕ)=tan2α
Hence,thelocusof(x1,y1)isx2−y2−2xycot2α=a2.
Exercises
1. Showthattheareaofthetrianglewithvertices(a,b),(x1,y1)and(x2,y2)wherea,x1andx2areingeometricprogressionwithcommonratiorandb,y1andy2areingeometricprogressionwith
commonratiosis
2. IfP(1,0),Q(−1,0)andR(2,0)arethreegivenpoints,thenshowthatthelocusofthepointS
satisfyingtherelationSQ2+SR2−2SP2isastraightlineparalleltothey-axis.
3. Showthatthepoints(p+1,1),(2p+1,3)and(2p+2,2)arecollinearifp=2or
4. Showthatthemidpointoftheverticesofaquadrilateralcoincideswiththemidpointoftheline
joiningthemidpointofthediagonals.
5. Showthatift1andt2aredistinctandnonzero,then and(0,0)arecollinear.
6. Ifthepoints arecollinearforthreedistinctvaluesa,
bandc,thenshowthatabc−(bc+ca+ab)+3(a+b+c)=0.7. PerpendicularstraightlinesaredrawnthroughthefixedpointC(a,a)tomeettheaxesofxandyatAandB.AnequilateraltriangleisdescribedwithABasthebaseofthetriangle.Provethatthe
equationofthelocusofCisthecurvey2=3(x2+a2).8. TheendsAandBofastraightlinesegmentofconstantlengthcslidesuponthefixedrectangular
axesOXandOY,respectively.IftherectangleOAPBiscompleted,thenshowthatthelocusofthe
footoftheperpendiculardrawnfromPtoABis .
9. ThepointAdividesthelinejoiningP(1,−5)andQ(3,5)intheratiok:1.FindthetwovaluesofkforwhichtheareaofthetriangleABCisequalto2unitsinmagnitudewhenthecoordinatesofBandCare(1,5)and(7,−2),respectively.
10. ThelinesegmentjoiningA(3,0)andB(0,2)isrotatedaboutapointAintheanticlockwisedirectionthroughanangleof45°andthusBmovestoC.IfpointDbethereflectionofCinthey-axis,findthecoordinatesofD.
Ans.:
11. If(a,b),(h,k)and(p,q)bethecoordinatesofthecircumcentre,thecentroidandtheorthocentreofatriangle,provethat3h=p+2α.
12. Provethatinarightangledtriangle,themidpointofthehypotenuseisequidistantfromitsvertices.
13. IfGisthecentroidofatriangleABC,thenprovethat3(GA2+GB2+GC2)=AB2+BC2+CA2.14. Showthatthelinejoiningthemidpointofanytwosidesofatriangleishalfofthethirdside.15. Provethatthelinejoiningthemidpointsoftheoppositesidesofaquadrilateralandthelinejoining
themidpointsofthediagonalsareconcurrent.
16. IfΔ1andΔ2denotetheareaofthetriangleswhoseverticesare(a,b),(b,c),(c,a)and(bc−a2,
ca−b2),(ca−b2,ab−c2)and(ab−c2,bc−a2),respectively,thenshowthatΔ2=(a+b+
c)2Δ1.17. Provethatiftwomediansofatriangleareequal,thetriangleisisosceles.
18. Ifa,bandcbethepth,qthandrthtermsofaHP,thenprovethatthepointshavingcoordinates(ab,r),(bc,p)and(ca,q)arecollinear.
19. Provethatapointcanbefoundthatisatthesamedistancefromeachofthefourpoints
20. If(x1,y1)(x2,y2)(x3,y3)and(x4,y4)betheverticesofaparallelogramandx1x3+y1y3=x2x1+y2y1thenprovethattheparallelogramisarectangle.
21. InanyΔABC,provethatAB2+AC2=2(AD2+DC2)whereDisthemidpointofBC.22. IfGisthecentroidofatriangleABCandObeanyotherpoint,thenprovethat
i. AB2+BC2+CA2=3(GA2+GB2+GC2)
ii. OA2+OB2+OC2=GA2+GB2+GC2+3GO2
23. Findtheincentreofthetrianglewhoseverticesare(20,7),(−36,7)and(0,−8).
Ans.:
24. IfA,BandCarethepoints(−1,5),(3,1)and(5,7),respectively,andD,EandFarethemidpointsofBC,CAandAB,respectively,provethatareaofΔABCisfourtimesthatofΔDEF.
25. IfD,EandFdividethesidesBC,CAandABofΔABCinthesameratio,provethatthecentroidofΔABCandΔDEFcoincide.
26. AandBarethefixedpoints(a,0)and(−a,0).FindthelocusofthepointPthatmovesinaplanesuchthat
i. PA2+PB2=2k2
ii. PA2−PB2=2PC2whereCisthepoint(c,0)
Ans.:(i)2ax+k2=0
(ii)2cx=c2−a2
27. If(xi,yi),i=1,2,3aretheverticesoftheΔABCanda,bandcarethelengthsofthesidesBC,CA
andAB,respectively,showthattheincentreofthetriangleABCis
28. Showthatthepoints(−a,−b),(0,0),(a,b)and(a2,b2)areeithercollinear,theverticesofaparallelogramortheverticesofarectangle.
29. ThecoordinatesofthreepointsO,AandBare(0,0),(0,4)and(6,0),respectively.ApointPmovessothattheareaofΔPOAisalwaystwicetheareaofΔPOB.FindtheequationofthelocusofP.
Ans.:x2−9y2=0
30. ThefourpointsA(x1,0),B(x2,0),C(x3,0)andD(x4,0)aresuchthatx1,x2aretherootsofthe
equationax2+2hx+b=0andx3,x4aretherootsoftheequationa1x2+2h1x+b1=0.Show
thatthesumoftheratiosinwhichCandDdivideABiszero,providedab1+a1b=2hh1.
31. If thenshowthatthetrianglewithvertices(xi,yi),i=1,2,3and(ai,bi),i=
1,2,3arecongruent.32. Thepoint(4,1)undergoesthefollowingthreetransformationssuccessively:
i. Reflectionabouttheliney=xii. Transformationthroughadistanceof2unitsalongthepositivedirectionofx-axis
iii. Rotationthroughanangleof abouttheoriginintheanticlockwisedirection.
Findthefinalpositionofthepoint.33. ShowthatthepointsP(2,−4),Q(4,−2)andR(1,1)lieonastraightline.Find(i)theratioPQ:QR
and(ii)thecoordinatesoftheharmonicconjugationofQwithrespecttoPandR.34. Ifapointmovessuchthattheareaofthetriangleformedbythatpointandthepoints(2,3)and
(−3,4)is8.5squareunits,showthatthelocusofthepointisx+5y−34=0.35. Showthattheareaofthetrianglewithvertices(p+5,p−4),(p−2,p+3)and(p,p)is
independentofp.
Chapter2
TheStraightLine
2.1INTRODUCTION
Inthepreviouschapter,wedefinedthatthelocusofapointisthepathtracedoutbyamovingpointaccordingtosomegeometricallaw.Weknowthatthelocusofapointwhichmovesinsuchawaythatitsdistancefromafixedpointisalwaysconstant.
2.1.1DeterminationoftheGeneralEquationofaStraightLine
SupposethepointP(x,y)movessuchthatP(x,y),A(4,−1),andB(2,3)forma
straightline.Then,
⇒x(−4)−y(2)+14=0
(i.e.)4x+2y−14=0or2x+y−7=0,whichisafirstdegreeequationinxandythatrepresentsastraightline.Thegeneralequationofastraightlineisax+by+c=0.Supposeax+by+c
=0isthelocusofapointP(x,y).IfthislocusisastraightlineandifP(x1,y1)andQ(x2,y2)beanytwopointsonthelocusthenthepointRwhichdividesPQwithratioλ:1isalsoapointontheline.SinceP(x1,y1)andQ(x2,y2)lieonthelocusax+by+c=0,
Onmultiplyingequation(2.2)byλandaddingwithequation(2.1),weget
λ(ax2+by2+c)+(ax1+by1+c)=0.(i.e.)a(λx2+x1)+b(λy2+y1)+c(λ+1)=0.
Ondividingbyλ+1,weget
Equation(2.3)showsthatthepoint liesonthelocusax+by
+c=0.ThisshowsthatthepointwhichdividesPQintheratioλ:1alsoliesonthelocuswhichisthedefinitionforastraightline.∴ax+by+c=0alwaysrepresentsastraightline.
Note2.1.1.1:Theaboveequationcanbewrittenintheform
whichisoftheformAx+By+1=0.Hence,therearetwoindependentconstantsinequationofastraightline.Now,welookintovariousspecialformsoftheequationofastraightline.
2.1.2EquationofaStraightLineParalleltoy-axisandataDistanceofhunitsfromx-axis
LetPQbethestraightlineparalleltoy-axisandataconstantdistancehunitsfromy-axis.TheneverypointonthelinePQhasthex-coordinateh.HencetheequationofthelinePQisx=h.
Note2.1.2.1:
1. Similarly,theequationofthelineparalleltox-axisandatadistancekfromitisy=k.2. Theequationofx-axisisy=0.3. Theequationofy-axisisx=0.
2.2SLOPEOFASTRAIGHTLINE
Ifastraightlinemakesanangleθwiththepositivedirectionofx-axisthentanθiscalledtheslopeofthestraightlineandisdenotedbym.
∴m=tanθ.
Wecannowdeterminetheslopeofastraightlineintermsofcoordinatesoftwopointsontheline.LetP(x1,y1)andQ(x2,y2)bethetwogivenpointsonaline.DrawPLandQMperpendicularstox-axis.LetPQmakeanangleθwithOX.
DrawQRperpendiculartoLP.Then .
2.3SLOPE-INTERCEPTFORMOFASTRAIGHTLINE
Findtheequationofthestraightline,whichmakesanangleθwithOXandcutsoffaninterceptconthey-axis.
LetP(x,y)beanypointonthestraightlinewhichmakesanangleθwithx-axis.
,OB=c=y-intercept.DrawPLperpendiculartox-axisandBN
perpendiculartoLP.Then, .BN=OL=x.
∴NP=LP−LN=LP−OB=y−c.
InΔNBP,
ThisequationistrueforallpositionsofPonthestraightline.Hence,thisistheequationoftherequiredline.
2.4INTERCEPTFORM
Findtheequationofthestraightline,whichcutsoffinterceptsaandb,respectivelyonxandyaxes.
LetP(x,y)beanypointonthestraightlinewhichmeetsxandyaxesatAandB,respectively.LetOA=a,OB=b,ON=x,andNP=y;NA=OA−ON=a−x.TrianglesPNAandBOAaresimilar.Therefore,
.ThisresultistrueforallpositionsofP
onthestraightlineandhencethisistheequationoftherequiredline.
2.5SLOPE-POINTFORM
Findtheequationofthestraightlinewithslopemandpassingthroughthegivenpoint(x1,y1).Theequationofthestraightlinewithagivenslopemis
Here,cisunknown.Thisstraightlinepassesthroughthepoint(x1,y1).Thepointhastosatisfytheequationy=mx+c.∴y1=mx1+c.Substitutingthevalueofcinequation(2.4),wegetthe
equationofthelineas
y=mx+y1−mx1⇒y−y1=m(x−x1).
2.6TWOPOINTSFORM
Findtheequationofthestraightlinepassingthroughtwogivenpoints(x1,y1)and(x2,y2).
where,misunknown.Theslopeofthestraightlinepassingthroughthepoints
Bysubstitutingequation(2.6)inequation(2.5),wegettherequiredstraightline
2.7NORMALFORM
Findtheequationofastraightlineintermsoftheperpendicularpfromtheorigintothelineandtheanglethattheperpendicularlinemakeswithaxis.
DrawOL⊥AB.LetOL=p.
Let
Therefore,theequationofthestraightlineABis
(i.e.)xcosα+ysinα=p
2.8PARAMETRICFORMANDDISTANCEFORM
Letastraightlinemakeanangleθwithx-axisandA(x1,y1)beapointontheline.DrawAL,PMperpendiculartox-axisandAQperpendiculartoPM.Then,
InΔPAQ,x−x1=rcosθ;y−y1=rsinθ.
Thesearetheparametricequationsofthegivenline.
Note2.8.1:Anypointonthelineisx=x1+rcosθ,y=y1+rsinθ.
Note2.8.2:risthedistanceofanypointonthelinefromthegivenpointA(x1,y1).
2.9PERPENDICULARDISTANCEONASTRAIGHTLINE
Findtheperpendiculardistancefromagivenpointtothelineax+by+c=0.
LetR(x1,y1)beagivenpointandax+by+c=0bethegivenline.ThroughRdrawthelinePQparalleltoAB.DrawOSperpendiculartoABmeetingPQatT.
LetOS=pandPT=p1.Let .ThentheequationofABis
whichisthesameas
Equations(2.7)and(2.8)representthesamelineand,therefore,identifying
weget
TheequationofthelinePQisxcosα+ysinα=p.SincethepointR(x1,y1)liesonthelinex1cosα+ysinα−p1=0.
∴p1=x1cosα+y1sinα.Then,thelengthoftheperpendicularlinefromRtoAB
Note2.9.1:Theperpendiculardistancefromtheoriginonthelineax+by+
2.10INTERSECTIONOFTWOSTRAIGHTLINES
Letthetwointersectingstraightlinesbea1x+b1y+c1=0anda2x+b2y+c2=0.Letthestraightlinesintersectatthepoint(x1,y1).Then(x1,y1)liesonboththelinesandhencesatisfytheseequations.Then
Solvingtheequations,weget
Therefore,thepointofintersectionis
Findtheratioatwhichthelineax+by+c=0dividesthelinejoiningthepoints(x1,y1)and(x2,y2).
Letthelineax+by=c=0dividethelinejoiningthepointsP(x1,y1)andQ(x2,y2)intheratioλ:1.Then,thecoordinatesofthepointofdivisionRare
Thispointliesonthelineax+by+c=0
Note2.10.1:
1. Ifλispositivethenthepoints(x1,y1)and(x2,y2)lieontheoppositesidesofthelineax+by+c=0.
2. Ifλisnegativethenthepoints(x1,y1)and(x2,y2)lieonthesamesideofthelineax+by+c=0.3. Inotherwords,iftheexpressionsax1+by1+candax2+by2+c2areofoppositesignsthenthe
point(x1,y1)and(x2,y2)lieontheoppositesidesofthelineax+by+c=0.Iftheyareofthesamesignthenthepoints(x1,y1)and(x2,y2)lieonthesamesideofthelineax+by+c=0.
Findtheequationofastraightlinepassingthroughintersectionofthelinesa1x+b1y+c=0anda2x+b2y+c=0.Considertheequation
Considertheequation
Thisisalinearequationinxandyandhencethisequationrepresentsastraightline.Let(x1,y1)bethepointofintersectionofthelinesa1x+b1y+c1=0anda2x+b2y+c2=0.Then(x1,y1)hastosatisfythetwoequations:
Onmultiplyingequation(2.11)byλandaddingwithequation(2.10)weget,
Thisequationshowsthatthepointx=x1andy=y1satisfiesequation(2.9).Hencethepoint(x1,y1)liesonthestraightlinegivenbytheequation(2.9),whichisalinepassingthroughtheintersectionofthelinesa1x+b1y+c1=0anda2x+b2y+c2=0.
2.11CONCURRENTSTRAIGHTLINES
Considerthreestraightlinesgivenbyequations:
Thepointofintersectionoflinesgivenbyequations(2.12)and(2.13)is
Ifthethreegivenlinesareconcurrent,theabovepointshouldlieonthestraightlinegivenbyequation(2.14).
Thisistherequiredconditionforthethreegivenlinestobeconcurrent.The
aboveconditioncanbeexpressedindeterminantform
Ifl,m,andnareconstantssuchthatl(a1x+b1y+c1)+m(a2x+b2y+c2)+n(a3x+b3y+c3)vanishesidenticallythenprovethatthelinesa1x+b1y+c1=0,a2x+b2y+c2=0,anda3x+b3y+c3=0areconcurrent.Letthelinesa1x+b1y+c1=0anda2x+b2y+c2=0meetatthepoint(x1,
y1).
Forallvaluesofxandygiventhat,
Thenitwillbetrueforx=x1andy=y1.∴l(a1x1+b1y1+c1)+m(a2x1+b2y1+c2)+n(a3x1+b3y1+c3)=0.
Usingequations(2.15)and(2.16),wegeta3x1+b3y1+c3=0.Thatis,thepoint(x1,y1)liesonthelinea3x+b3y+c3=0.Therefore,thelinesa1x+b1y+c1=0,a2x+b2y+c2=0,a3x+b3y+c3=0
areconcurrentat(x1,y1).
2.12ANGLEBETWEENTWOSTRAIGHTLINES
Letθbetheanglebetweentwostraightlines,whoseslopesarem1andm2.Letthetwolineswithslopesm1andm2makeanglesθ1andθ2withx-axis.Then,m1=tanθ1,m2=tanθ2.Also,θ=θ1−θ2
IftheRHSispositive,thenθistheacuteanglebetweenthelines.IfRHSisnegative,thenθistheobtuseanglebetweenthelines.
Note2.12.1:Ifthelinesareparallelthenθ=0andtanθ=tan0=0.
Note2.12.2:Ifthelinesareperpendicularthen,
Therefore,
1. Iftwolinesareparallelthentheirslopesareequal.2. Ifthetwolinesareperpendicularthentheproductoftheirslopesis−1.
2.13EQUATIONSOFBISECTORSOFTHEANGLEBETWEENTWOLINES
LetABandCDbethetwointersectingstraightlinesintersectingatP.Lettheselinesberepresentedbytheequationsa1x+b1y+c1=0anda2x+b2y+c2=0.
LetPRandPR′bethebisectorsofangles and ,respectively.Thenthe
perpendiculardistancesfromR(orR′)AB,andCDareequal.
Ifc1andc2arepositive,thentheequationsofthebisectorcontainingtheoriginisgivenby
Theequationofthebisectornotcontainingtheoriginis
Ifc1andc2arenotpositivethentheequationsoflinesshouldbewritteninsuchawaythatc1andc2arepositive.
Note2.13.1:Wecaneasilyobservethatthetwobisectorsareatrightangles.
ILLUSTRATIVEEXAMPLES
Example2.1
Findtheequationofthestraightlinewhichisatadistanceof10unitsfromx-axis.
Solution
Theequationoftherequiredstraightlineisx=10orx−10=0.
Example2.2
Findtheequationofthestraightlinewhichisatadistanceof−15unitsfromy-axis.
Solution
Theequationoftherequiredlineisy=−15ory+15=0.
Example2.3
Findtheslopeofthelinejoiningthepoints(2,3)and(4,−5).
Solution
Theslopeofthelinejoiningthetwogivenpoints(x1,y1)and(x2,y2)is
Therefore,theslopeofthelinejoiningthetwogivenpointsis
Example2.4
Findtheslopeoftheline2x−3y+7=0.
Solution
Theequationofthelineis2x−3y+7=0(i.e.)3y=2x+7.
Therefore,slopeoftheline= .
Example2.5
Findtheequationofthestraightlinemakinganangle135°withthepositivedirectionofx-axisandcuttingofanintercept5onthey-axis.
Solution
Theslopeofthestraightlineis
yintercept=c=5.Therefore,theequationofthestraightlineis
Example2.6
Findtheequationofthestraightlinecuttingofftheintercepts2and−5ontheaxes.
Solution
Theequationofthestraightlineis .Here,a=2andb=−5.
Therefore,theequationofthestraightlineis or5x−2y=10.
Example2.7
Findtheequationofthestraightlinepassingthroughthepoints(7,−3)andcuttingoffequalinterceptsontheaxes.
Solution
Lettheequationofthestraightlinebe
(i.e.)x+y=a.
Thisstraightlinepassesthroughthepoint(7,−3).Therefore,7−3=a(i.e.)a=4.∴Theequationofthestraightlineisx+y=4.
Example2.8
Findtheequationofthestraightline,theportionofwhichbetweentheaxesisbisectedatthepoint(2,−5).
Solution
Lettheequationofthestraightlinebe
LetthelinemeetthexandyaxesatAandB,respectively.Thenthecoordinates
ofAandBare(a,0)and(0,b).ThemidpointofABis .However,the
midpointisgivenas(2,−5).
Therefore,
∴a=4andb=−10.
Hence,theequationofthestraightlineis
(i.e.)5x−2y=20.
Example2.9
Findtheequationofthestraightlineoftheportionofwhichbetweentheaxesisdividedbythepoint(4,3)intheratio2:3.
Solution
Lettheequationofthestraightlinebe
LetthislinemeetthexandyaxesatAandB,respectively.ThecoordinatesofAandBare(a,0)and(0,b),respectively.ThecoordinatesofthepointthatdividesABintheratio2:3are
Thispointisgivenas(4,3).
Therefore,
∴Theequationofthestraightlineis (i.e.)9x+8y=60.
Example2.10
Findtheequationstothestraightlineseachofwhichpassesthroughthepoint(3,2)andintersectthexandyaxesatAandBsuchthatOA−OB=2.
Solution
Lettheequationofthestraightlinebe .Thisstraightlinepassesthrough
thepoint(3,2).
Also,giventhatOA−OB=2
Therefore,b=a−2.Substitutingthisinequation(2.20)weget3(a−2)+2a=a(a−2).
∴Thetwostraightlinesare and
(i.e.)x−y=1and2x+3y=12.
Example2.11
ShowthatthepointsA(l,1),B(5,−9),andC(−l,6)arecollinear.
Solution
TheslopeofABis
SincetheslopesofABandBCareequalandBisthecommonpoint,thepointsarecollinear.
Example2.12
Provethatthetrianglewhoseverticesare(−2,5),(3,−4),and(7,10)isarightangledisoscelestriangle.Findtheequationofthehypotenuse.
SolutionLetthepointsbeA(−2,5),B(3,−4),andC(7,10).AB2=(−2−3)2+(5+4)2=25+81=106,BC2=(3−7)2+(−4−10)2=16+196=212.AC2=(−2−7)2+(5−10)2=81+25=106.Therefore,AB2+AC2=BC2
andAB=AC.Hence,the∆ABCisarightangledisoscelestriangle.TheequationofthehypotenuseBCis
Example2.13
Findtheequationofthestraightlinewhichcutsoffinterceptsontheaxesequalinmagnitudebutoppositeinsignandpassingthroughthepoint(4,7).
Solution
Lettheequationofthestraightlinecuttingoffinterceptsequalinmagnitudebut
oppositeinsignbe (i.e.)x−y=a.
Thispassesthroughthepoint(4,7).Therefore,4−7=a(i.e.)a=−3.Hence,theequationofthestraightlineisx−y+3=0.
Example2.14
Findtheratioinwhichtheline3x−2y+5=0dividesthelinejoiningthepoints(6,−7)and(−2,3).
Solution
Lettheline3x−2y+5=0dividethelinejoiningthepointsA(6,−7)andB(−2,
3)intheratiok:1.Thenthecoordinatesofthepointofdivisionare
.Sincethispointliesonthestraightline3x−2y+5=0,weget
∴Therequiredratiois37:7.
Example2.15
Provethatthelines3x−4y+5=0,7x-8y+5=0,and4x+5y=45areconcurrent.
Solution
Given
Solvingequations(2.22)and(2.23),wegetthepointofintersectionofthetwolines.
∴Fromequation(2.22),15−4y=−5.∴y=5.Hence,thepointofintersectionofthelinesis(5,5).Substitutingx=5andy=
5,inequation(2.24),weget20+25=45whichistrue.∴Thethirdlinealsopassesthroughthepoints(5,5).Henceitisprovedthat
thethreelinesareconcurrent.
Example2.16
Findthevalueofasothatthelinesx−6y+a=0,2x+3y+4=0,andx+4y+1=0areconcurrent.
Solution
Given
Solvingtheequations(2.26)and(2.27)weget,
Onsubtracting,weget5y=2
∴Fromequation(2.27)
Hence,thepointofintersectionofthelinesis .Sincethelinesare
concurrentthispointshouldlieonx−6y+a=0.
Example2.17
Provethatforallvaluesofλthestraightlinex(2+3λ)+y(3−λ)−5−2λ=0passesthroughafixedpoint.Findthecoordinatesofthefixedpoint.
Solution
x(2+3λ)+y(3−λ)−5−2λ=0.Thisequationcanbewrittenintheform
Thisequationrepresentsastraightlinepassingthroughtheintersectionoflines
forallvaluesofλ.
Onadding,weget11x=11⇒x=1andhencefromequation(2.29)wegety=1.Therefore,thepointofintersectionofstraightlines(2.29)and(2.30)is(1,1).
Thestraightline(2.28)passesthroughthepoint(1,1)forallvaluesofλ.Hence(2.28)passesthroughthefixedpoint(1,1).
Example2.18
Findtheequationofthestraightlinepassingthroughtheintersectionofthelines3x−y=5and2x+3y=7andmakinganangleof45°withthepositivedirectionofx-axis.
SolutionSolvingtheequations,
Weget,
Onadding,weget11x=22.
∴x=2.Fromequation(2.31),6−y=5.
∴y=1.Hence(2,1)isthepointofintersectionofthelines(2.31)and(2.32).
Theslopeoftherequiredlineism=tanθ,m=tan45°=1.Therefore,theequationoftherequiredlineisy−y=m(x−x1)(i.e.)y−1=1(x−2)⇒x−y=1.
Example2.19
Findtheequationofthestraightlinepassingthroughtheintersectionofthelines7x+3y=7and2x+y=2andcuttingoffequalinterceptsontheaxes.
Solution
Thepointofintersectionofthelinesisobtainedbysolvingthefollowingtwoequations:
Onsubtracting,wegetx=1andhencey=0.Therefore,thepointofintersection
is(1,0).Theequationofthestraightlinecuttingoffequalinterceptsis
(i.e)x+y=a.Thisstraightlinepassesthrough(1,0).Therefore,1+0=a(i.e.)a=1.
Hence,theequationoftherequiredstraightlineisx+y=1.
Example2.20
Findtheequationofthestraightlineconcurrentwiththelines2x+3y=3andx+2y=2andalsoconcurrentwiththelines3x−y=1andx+5y=11.
Solution
Thepointofintersectionofthelines2x+3y=3andx+2y=2isobtainedbysolvingthefollowingtwoequations:
Onsubtracting,wegety=1andhencex=0.Therefore,thepointofintersectionis(0,1).
Onadding,weget16x=16whichgivesx=1andhencey=2.Thepointofintersectionofthesecondpairoflinesis(1,2).Theequationofthelinejoiningthetwopoints(0,1)and(1,2)is
Example2.21
Findtheanglebetweenthelines
Solution
Theslopeoftheline .Therefore, (i.e.)θ1=
60°.Theslopetheline Therefore,
.Theanglebetweenthelinesisθ1−θ2=30°.
Example2.22
Findtheequationoftheperpendicularbisectorofthelinejoiningthepoints(−2,6)and(4,−6).
Solution
Theslopeofthelinejoiningthepoints(−2,6)and(4,−6)is .
Therefore,theslopeoftheperpendicularlineis .Themidpointoftheline
joiningthepoints(−2,6)and(4,−6)is (i.e.)(1,0).
Therefore,theequationoftheperpendicularbisectorisy−y1=m(x−x1)
(i.e.)y−0= (x−1)⇒2y=x−1orx−2y−1=0.
Example2.23
A(4,1),B(7,4),andC(5,−2)aretheverticesofatriangle.FindtheequationoftheperpendicularlinefromAtoBC.
Solution
TheslopeofthelineBCis
Therefore,theslopeoftheperpendicularADtoBCis− .Hence,theequationof
theperpendicularfromA(4,1)onBCisy−y1=m(x−x1)
Example2.24
Thefootoftheperpendicularfromthepoint(1,2)onalineis(3,–4).Findtheequationoftheline.
Solution
LetABbethelineandD(3,−4)bethefootoftheperpendicularfromC(1,2)
TheslopeofthelineCDis
Therefore,theslopeofthelineABis .
TheequationofthelineABisy−y1=m(x−x1)
Example2.25
Findtheequationoftherightbisectorofthelinejoiningthepoints(2,3)and(4,5).
Solution
Therightbisectoristheperpendicularbisectorofthelinejoiningthepoints(2,
3)and(4,5).Themidpointofthelineis
Therefore,theslopeofthegivenlineis
∴Theslopeoftherightbisectoris−1.Theequationoftherightbisectorisy−y1=m(x−x1)⇒y−4=−1(x−3)ory−4=−x+3orx+y=7.
Example2.26
Findthepointontheline3y−4x+11=0whichisequidistantfromthepoints(3,2)and(−2,3).
Solution
LetP(x1,y1)bethepointontheline3y−4x+11=0whichisequidistantfromthepointsA(3,2)andB(−2,3).
Since,
Sincethe(x1,y1)liesontheline,
Substitutingy1=5x1in(2.40),weget15x1−4x1+11=0.∴x1=−1andhencey1=−5.Therefore,therequiredpointis(−1,−5).
Example2.27
Findtheequationofthelinepassingthroughthepoint(2,3)andparallelto3x−4y+5=0.
Solution
Theslopeoftheline3x−4y+5=0is .
Therefore,theslopeoftheparallellineisalso .
Hencetheequationoftheparallellinethrough(2,3)isy−y1=m(x−x1)
Example2.28
Findtheequationofthelinepassingthroughthepoint(4,−5)andisperpendiculartotheline7x+2y=15.
Solution
Theslopeoftheline
Therefore,theslopeoftheperpendicularlineis .Theequationofthe
perpendicularlinethrough(4,−5)isy−y1=m(x−x1)
Example2.29
Findtheequationofthelinethroughtheintersectionof2x+y=8and3x+7=2yandparallelto4x+y=11.
Solution
Thepointofintersectionofthelines2x+y=8and3x+7=2yisobtainedbysolvingthefollowingtwoequations:
Onadding,weget
Therefore,thepointofintersectionis
Theslopeoftheline4x+y=11is−4.Theslopeoftheparallellineisalso−4.Theequationoftheparallellineis
(i.e.)28x+7y=74.
Example2.30
Findtheimageoftheoriginontheline3x−2y=13.
Solution
LetO′(x1,y1)betheimageofOonthelineAB.ThenCisthemidpointofOO′.
TheslopeofthelineOO′is
TheequationofthelineOO′is
Solvingtheequations
TogetthecoordinatesofC:
Therefore,Cis(3,−2).CbeingthemidpointofOO′.
Therefore,theimageis(6,−4).
Example2.31
Findtheequationofthestraightlinepassingthroughtheintersectionofthelines3x+4y=17and4x−2y=8andperpendicularto7x+5y=12.
Solution
(2.45)×1+(2.46)×2gives
From(2.45),9+4y=17.Therefore,y=2.Hence(3,2)isthepointof
intersectionofthelines(2.45)and(2.46).Theslopeoftheline
Therefore,theslopeoftheperpendicularlineis
7y−14=5x−15or5x−7y=1.
Example2.32
Findtheorthocentreofthetrianglewhoseverticesare(5,−2),(−1,2),and(1,4).
Solution
Slopeof Therefore,slopeoftheperpendicularADis−1.
TheequationofthelineADisy+2=−1(x−5).
Slopeof Therefore,slopeofBEis
TheequationofBEis
Solvingtheequations(2.47)and(2.48),wegetthecoordinatesoftheorthocentre:
Onadding5x=1orx= .
From(2.47),
∴Theorthocentreis
Example2.33
Thepoints(1,3)and(5,1)aretwooppositeverticesofarectangle.Theothertwoverticeslieontheliney=2x+c.Findcandtheremainingtwovertices.
Solution
LetABCDbetherectanglewithAandCasthepointswithcoordinates(1,3)and(5,1),respectively.Inarectanglethediagonalsbisecteachother.
ThemidpointACis
AsthispointliesonBDwhoseequationisy=2x+c.Weget2=6+corc=−4.Therefore,theequationofthelineBDisy=2x−4.Therefore,thecoordinatesofanypointonthislineis(x,2x−4).IfthisisthepointBthenAB2+BC2=AC2.
Asy=2x−4,thecorrespondingvaluesofy=0,4.Therefore,thecoordinatesofBandDare(2,0)and(4,4).
Example2.34
Ifa,b,andcaredistinctnumbersdifferentfrom1thenshowthatthepoints
arecollinearifab+bc+ca−abc=
3(a+b+c).
Solution
LetA,B,andClieonthestraightlinepx+qy+r=0.
Thentheequationofthelinesatisfiesthecondition
wheret=a,b,andc(i.e.)pt3+qt2+rt−3q−r=0.Here,a,b,andcaretherootsofthisequation.
Example2.35
Avertexofanequilateraltriangleisat(2,3)andtheequationoftheoppositesideisx+y=2.Findtheequationsoftheothersides.
Solution
TheslopeofBCis−1.LetmbeslopeofABorAc.Then
Therefore,theequationofothertwosidesare and
.
Example2.36
Onediagonalofasquareistheportionoftheline interceptedbetween
theaxes.Findtheequationoftheotherdiagonal.
Solution
TheslopeofABis LetmbeslopeofAC.
TheequationofthesideOCis
TheequationofthesideBDis
Theequationoftheotherdiagonalis
Example2.37
IftheverticesofΔABCare(xi,yi)i=1,2,3.Showthattheequationofthe
medianthroughAisgivenby
Solution
ThecoordinatesofthemidpointofBCare
TheequationofthemedianADisgivenby Sincetheareaof
alineiszero.
Example2.38
If(x,y)isanarbitrarypointontheinternalbisectorofverticalangleAofΔABC,where(xi,yi),i=1,2,3aretheverticesofA,B,andC,respectively,anda,b,andcarethelengthofthesidesBC,CA,andAB,respectively,provethat
Solution
InΔABC,ADistheinternalbisectorof .Weknowthat
ThecoordinatesofDare
TheequationofADisgivenby
Example2.39
Findtheorthocentreofthetrianglewhoseverticesare(a,0),(0,b),and(0,0).
Solution
Theorthocentreisthepointofconcurrenceofaltitudes.SinceOAandOBareperpendiculartoeachother,OAandOBarethealtitudesthroughAandBofΔABC.Therefore,0istheorthocentre.Hence,thecoordinatesoftheorthocentreis(0,0).
Example2.40
Provethattheorthocentreofthetriangleformedbythethreelines
liesonthelinex+a=0.
Solution
Theequationofthelinepassingthroughtheintersectionofthelines
TheslopeofthelineADis TheslopeofthelineBCist1.SinceADis
perpendiculartoBC,
TheequationofthelineADis
SimilarlytheequationofthelineBEis
Subtractingequations(2.52)from(2.53),t3(t1−t2)x+at3(t1−t2)=0.Sincet1≠t2,x+a=0theorthocentreliesonthelinex+a=0.
Example2.41
Showthatthereflectionofthelinepx+qy+r=0,onthelinelx+my+n=0is(px+qy+r)(l2+m2)−2(lp+mq)(lx+my+n)=0.
Solution
LetADbethereflectionofthelinepx+qy+r=0inthelinelx+my+n=0.ThentheequationoflineADispx+qy+r+k(lx+my+n)=0.ThentheperpendicularfromanypointonACtoABandADareequal.
SincethepointPliesonAC,lx+my+n=0,lx1+my1+n=0
HencetheequationofthelineADis(l2+m2)(px+qy+r)−2(pl+qm)(lx+my+n)=0.
Example2.42
Thediagonalsoftheparallelogramaregivenbythesidesu=p,u=q,v=r,v=swhereu=ax+by+candv=a1x+b1y+c1.Showthattheequationofthediagonalwhichpassesthroughthepointsofintersectionofu=p,v=randu=
qandr=sisgivenby
Solution
Consider
Thisisalinearequationinxandyand,therefore,itrepresentsastraightline.ThecoordinatesofBaregivenbytheintersectionofthelinesu=pandv=s.However,u=pandv=rsatisfiestheequation(2.54).Inaddition,u=q,andv=ssatisfytheequation(2.54)andhencetheequation(2.54)isthelinepassingthroughBandDandrepresentstheequationofthediagonalBD.
Example2.43
AlinethroughthepointA(−5,−4)meetsthelinesx+3y+2=0,2x+y+4=0,
andx−y−5=0atthepointsB,C,andD,respectively.If
findtheequation.
Solution
Theequationofthelinepassingthroughthepoint(−5,−4)is
Anypointonthelineis(rcosθ−5,rsinθ−4).Thepointmeetsthelinex+3y+2=0atBthenAB=r·(rcosθ−5)+3(rsinθ−4)+2=0.
Iftheline(2.55)meetstheline2x+y+4=0atCthen2(rcosθ−5)+(rsinθ−4)+4=0.Now,r=ACand
Iftheline(2.55)meetsthelinex−y−5=0then(rcosθ−5)−(rsinθ−4)−5=0andhereAD=r.
Giventhat
Hence,theequationofthelineis
Example2.44
AvariablestraightlineisdrawnthroughOtocuttwofixedlinesL1andL2atA1
andA2.ApointAistakenonthevariablelinesuchthat Showthat
thelocusofPisastraightlinepassingthroughthepointofintersectionofL1andL2.
Solution
LettheequationofthelineOXbe whereOA=r.Anypointonthis
lineis(rcosθ,rsinθ).LetOA1=r1andOA2=r2.A1is(r1cosθ,r1sinθ)andA2is(r2cosθ,r2sinθ).LetthetwofixedstraightlinesbeL1:l1x+m1y−1=0andL2:l2x+m2y−1=0.SincethepointsA1andA2lieonthetwolines,respectively,
(i.e.)p(l1x+m1y−1)+q(l2x+m2y−1)=0whichisastraightlinepassingthroughthepointofintersectionofthetwofixedstraightlinesL1=0andL2=0.
Example2.45
Iftheimageofthepoint(x1,y1)withrespecttothelinemy+lx+n=0isthe
point(x2,y2)showthat
Solution
Q(x2,y2)isthereflectionofP(x1,y1)onthelinelx+my+n=0.Themidpoint
ofPQliesonthelinelx+my+n=0.TheslopeofPQis Theslopeofthe
lineislx+my+n=0is Sincethesetwolinesareperpendicular,
Example2.46
ProvethattheareaofthetrianglewhoserootsareLr=arx+bry+cr(r=1,2,3)
is whereCiisthecofactorofci(i=1,2,3)inAgivenby
Solution
LetAr,Br,andCrbethecofactorsofar,br,andcrinD.Thepointofintersection
ofthelinesa1x+b1y+c1=0,a2x+b2y+c2=0is
∴Theverticesofthetriangleare Thentheareaof
thetriangleisgivenby,
whereDisthedeterminantformedbythecofactors.
Example2.47
AstraightlineLintersectsthesidesBC,CA,andABofatriangleABCinD,E,
andF,respectively.Showthat
Solution
LetDEFbethestraightlinemeetingBC,CA,andABatD,E,andF,respectively.LettheequationsofthelineDEFbelx+my+n=0.
LetDdivideBCintheratioλ:1.ThenthecoordinatesofDare
Asthispointliesonthelinelx+my+n=0.
Similarly
Multiplyingthesethreeweget
Example2.48
Astraightlineissuchthatthealgebraicsumofperpendicularsdrawnuponitfromanynumberoffixedpointsiszero.Showthatthestraightlinepassesthroughafixedpoint.
Solution
Let(x1,y1),(x2,y2),…,(xn,yn),benfixedpointsandax+by+c=0beagivenline.Thealgebraicsumoftheperpendicularsfrom(xi,yi),i=1,2,...,ntothislineiszero.
Thisequationshowsthatthepoint liesonthelineax
+by+c=0.Therefore,thelinepassesthroughafixedpoint.
Example2.49
Determineallthevaluesofαforwhichthepoint(α,α2)liesinsidethetriangleformedbythelines2x+3y−1=0,x+2y−3=0,and5x−6y−1=0.
Solution
∴L1(α,α2)=2a+3α2−1>0ifpointsAand(a,α2)liesonthesamesideoftheline.3α2+2a−1>0⇒(3α−1)(α+1)>0.
FromtheconditionsI,II,andIII,wehave
Example2.50
Findthedirectioninwhichastraightlinemustbedrawnthroughthepoint(1,4)sothatitspointofintersectionwiththelinex+y+5=0maybeatadistance
units.
Solution
Lettheequationofthelinethroughthepoint(1,4)be
Anypointonthislineis(rcosθ+1,rsinθ+4).Ifthispointliesonthelinex+y+5=0thenrcosθ+1+rsinθ+4+5=0.
∴Therequiredstraightlinemakesanangleof withthepositivedirectionsof
x-axisandpassesthroughthepoint(1,4).
Exercises
1. Findtheareaoftriangleformedbytheaxes,thestraightlineLpassingthroughthepoints(1,1)
and(2,0)andthelineperpendiculartotheLandpassingthrough
Ans.:
2. Theline3x+2y=24meetsy-axisatAandx-axisatB.TheperpendicularbisectorofABmeetsthelinethrough(0,−1)paralleltox-axisatC.FindtheareaΔABC.
Ans.:91sq.units
3. If(x,y)beanarbitrarypointonthealtitudethroughAofΔABCwithvertices(xi,yi),i=1,2,3
thentheequationofthealtitudethroughAis
4. Arayoflightissentalongthelinex−2y−3=0.Uponreachingtheline3x−2y−5=0therayisreflectedfromit.Findtheequationofthelinecontainingthereflectedray.
Ans.:29x−2y−31=0
5. Theextremitiesofthediagonalsofasquareare(1,1)and(−2,−1).Obtaintheequationoftheotherdiagonal.
Ans.:6x+4y+3=0
6. Thestraightline3x+4y=5and4x−3y=15intersectatthepointA.Onthisline,thepointsBandCarechosensothatAO=AC.FindthepossibleequationsofthelineBCpassingthroughthepoint(1,2).
Ans.:x−7y+13=0and7x+y−9=0
7. Theconsecutivesidesofaparallelogramare4x+5y=0and7x+2y=0.Iftheequationofonediagonalis11x+7y=9,findtheequationoftheotherdiagonal.
Ans.:x−y=0
8. Showthatthelinesax±by±c=0enclosearhombusofarea
9. IftheverticesofaΔOBCareO(0,0),B(−3,−1),andC(−1,−3),findtheequationoftheline
paralleltoBCandintersectingsidesOBandOCwhoseperpendiculardistancefrom(0,0)is .
Ans.:2x+2y+ =0
10. Findthelocusofthefootoftheperpendicularfromtheoriginuponthelinejoiningthepoints(acosθ,bsinθ)and(−asinθ,bcosθ)whereaisavariable.
Ans.:a2x2+b2y2=2(x2+y2)2
11. Showthatthelocusgivenbyx+y=0,(a-b)x+(a+b)y=2aband(a+b)x+(a−b)y=2ab
formanisoscelestrianglewhoseverticalangleis Determinethecentroidofatriangle.
Ans.:
12. Thesidesofaquadrilateralhavetheequations,x+2y=3,x=1,x−3y=4,and5x+y+12=0.Showthatthediagonalsofthequadrilateralareatrightangles.
13. GivennstraightlinesandafixedpointO.ThroughOastraightlineisdrawnmeetingtheselines
inthepointA1,A2,…,AnandapointAsuchthat Provethatthelocus
ofthepointAisastraightline.14. Findtheequationofthelinejoiningthepoint(3,5)tothepointofintersectionofthelines4x+y−
1=0and7x−3y−35=0andprovethatthelineisequidistantfromtheoriginandthepointsA,B,C,andD.
15. Findtheequationofthelinepassingthroughthepoint(2,3)andmakinginterceptsoflength2unitsandbetweenthelines.
Ans.:3x+4y−8=0andx−2=0
16. Ifxcosα+ysinα=pwhere beastraightline,provethattheperpendicularsp1,p2,and
p3onthelinefromthepoint(m2,2m),(mm′,m+m′),and(m′2,2m′),respectively,areinG.P.
17. Provethatthepoints(a,b),(c,d),and(a−c,b−d)arecollinearif(ad=bc).Also,showthatthestraightlinepassingthroughthesepointspassesthroughtheorigin.
18. Onediagonalofasquareisalongtheline8x−15y=0andoneofitsverticesis(1,2).Findtheequationsofthesidesofthesquarethroughthisvertex.
Ans.:2x+y=4andx−2y+3=0
19. Findtheorthocentreofatriangleformedbylineswhoseequationsarex+y=1,2x+8y=6,and4x−y+4=0.
Ans.:
20. Thesidesofatriangleareur=xcosαr+ysinαr−pr=0,r=1,2,3.Showthatitsorthocentreisgivenbyu1cos(α2−α3)=u2cos(α3−α1)=u3cos(α1−α2).
21. Findtheequationofstraightlinespassingthroughthepoint(2,3)andhavinganinterceptoflength2unitsbetweenthestraightlines2x+3y=3and2x+y=5.
Ans.:x=2,3x+4y=18
22. LetalineLhasinterceptsaandbonthecoordinateaxes.Whentheaxesarerotatedthroughanangle,keepingtheoriginfixed,thesamelineLhasinterceptspandq.Obtaintherelationbetweena,b,p,andq.
Ans.:
23. AlinethroughthevariablepointA(k+1,2k)meetstheline7x+y−16=0,5x−y+8=0,x−5y+8=0atB,C,andD,respectively.ProvethatAC,AB,andADareinG.P.
24. Findtheequationofthestraightlinespassingthrough(−2,−7)andhavinganinterceptoflength3betweenthestraightlines4x+3y=12and4x+3y=3.
Ans.:x+2=0,7x−24y+182=0
25. Alineissuchthatitssegmentbetweenthestraightlines5x−y−4=0and3x+4y−4=0isbisectedatthepoint(1,5).Obtainitsequation.
Ans.:83x−35y+92=0.
26. Provethatthe(a−b)x+(b−c)y+(c−a)=0,(a−c)x+(c−a)y+(a−b)=0,and(c−a)x+(a−b)y+(b−c)=0areconcurrent.
27. Twoverticesofatriangleare(5,−1)and(−2,3).Iftheorthocentreofthetriangleisattheorigin,findthecoordinatesofthethirdvertex.
Ans.:(−4,−7)
28. Alineintersectsx-axisatA(7,0)andy-axisatB(0,−5).AvariablelinePQwhichisperpendiculartoABintersectsx-axisatPandy-axisatQ.IfAQandBPintersectatR,thenfindthelocusofR.
Ans.:x2+y2−7x+5y=0
29. ArectanglePQRShasitssidePQparalleltotheliney=mxandverticesP,Q,andSonthelinesy=a,x=b,andx=−b,respectively.FindthelocusofthevertexR.
Ans.:(m2−1)x−my+b(m2+1)+am=0
30. Determinetheconditiontobeimposedonβsothat(O,β)shouldbeonorinsidethetrianglehavingsidesy+3x+2=0,3y−2x−5=0,and4y+x−14=0.
Ans.:
31. Showthatthestraightlines7x−2y+10=0,7x+2y−10=0,andy=2formanisoscelestriangleandfinditsarea.
Ans.:14sq.units
32. TheequationsofthesidesBC,CA,andABofatriangleABCareKr=arx+bry+cr=0,r=1,2,3.ProvethattheequationofalinedrawnthroughAparalleltoBCisK3(a2b1−a1b2)=K2(a3b1−a1b3).
33. ThesidesofatriangleABCaredeterminedbytheequationur=arx+bry+cr=0,r=1,2,3.ShowthatthecoordinatesoftheorthocentreofthetriangleABCsatisfytheequationλ1u1=λ2u2+λ3u3whereλ1=a2a3+b2b3,λ2=a3a1+b3b1,andλ3=a1a2+b1b2.
34. ProvethatthetwolinescanbedrawnthroughthepointP(P,Q)sothattheirperpendiculardistancesfromthepointQ(2a,2a)willbeequaltoaandfindtheirequations.
Ans.:y=a,4x−3y+3a=0.
35. Findthelocusofapointwhichmovessuchthatthesquareofitsdistancefromthebaseofanisoscelestriangleisequaltotherectangleunderitsdistancesfromtheothersides.
Ans.:
36. Provethatthelinesgivenby(b+c)x−bcy=a(b2+bc+c2),(c+a)x−cay=b(c2+ca+a2),
and(a+b)x−aby=c(a2+ab+b2)areconcurrent.37. Showthattheareaofthetriangleformedbythelinesy=m1x+c1,y=m2x+c2,andy=m3x+
c3is
38. Findthebisectoroftheacuteanglebetweenthelines3x+4y=1whichisthebisectorcontainingtheorigin.
Ans.:11x+3y−17=0(originliesintheobtuseanglebetweenthelines.)
39. Ifa1a2+b1b2>0provethattheoriginliesattheobtuseanglebetweenthelinesa1x+b1y+c1=0anda2x+b2y+c2=0,wherec1andc2bothbeingofthesamesign.
40. Findtheequationtothediagonalsoftheparallelogramformedbythelinesax+by+c=0,ax+by
+d=0,a′x+b′y+c′=0,a′x+b′y−d′=0.Showthattheparallelogramwillbearhombusif(a2
+b2)(c′−d′)2=(a′2+b′2)(c−d)2.41. AvariablelineisataconstantdistancepfromtheoriginandmeetscoordinateaxesinAandB.
ShowthatthelocusofthecentroidoftheΔOABisx−2+y−2=p−2.42. Amovinglineislx+my+n=0wherel,m,andnareconnectedbytherelational+bm+cn=0,
anda,b,andcareconstants.Showthatthelinepassesthroughafixedpoint.43. Findtheequationofbisectorofacuteanglebetweenthelines3x−4y+7=0and12x+5y−2=0.44. Qisanypointonthelinex−a=0andOistheorigin.IfAisthepoint(a,0)andQR,thebisector
meetsx-axisonR.ShowthatthelocusofthefootoftheperpendicularfromRtoOQisthe
(x−2a)(x2+y2+a2x)=0.45. Thelinesax+by+c=0,bx+cy+a=0,andcx+ay+b=0areconcurrentwherea,b,andcare
thesidesoftheΔABCinusualnotationandprovethatsin3A+sin3B+sin3C=3sinAsinBsinC.
46. AvariablestraightlineOPQpassesthroughthefixedpointO,meetingthetwofixedlinesinpointsPandQ.InthestraightlineOPQ,apointRistakensuchthatOP,OR,andOQareinharmonicprogression.ShowthatthelocusofpointQisastraightline.
47. Arayoflightissetalongthelinex−2y−3=0.Onreachingtheline3x−2y−5=0,therayisreflectedfromit.Findtheequationofthelinecontainingthereflectedray2qx−2y−31=0.
Ans.:2qx−2y−31=0.
48. LetΔABCbeatrianglewithAB=AC.IfDisthemidpointofBC,andEisthefootoftheperpendiculardrawnfromDtoACandFisthemidpointofBE.ProvethatAFisperpendiculartoBE.
Ans.:14x+23y−40=0.
49. TheperpendicularbisectorsofthesidesABandACofatriangleABCarex−y+5=0andx+2y=0,respectively.IfthepointAis(1,−2),findtheequationoftheline14x+23y−40=0.
50. Atriangleisformedbythelinesax+by+c=0,lx+my+n=0,andpx+qy+r=0.Showthat
thestraightline passesthroughtheorthocentreofthetriangle.
51. Provethatthediagonalsoftheparallelogramformedbythelinesax+by+c=0,ax+by+c′=
0,a′x+b′y+c=0,anda′x+b′y+c′=0willbeatrightanglesifa2+b2=a′2+b′2.
52. Onediagonalofasquareistheportionoftheline interceptedbetweentheaxes.Show
thattheextremitiesoftheotherdiagonalare
53. Showthattheoriginliesinsideatrianglewhoseverticesaregivenbytheequations7x−5y−11=0,8x+3y+31=0,andx+3y−19=0.
54. ArayoflighttravellingalongthelineOA,Obeingtheorigin,isreflectedbythelinemirrorx−y+1=0,thepointofincidenceAis(1,2).Thereflectedrayisagainreflectedbythemirrorx−y=1,thepointofincidencebeingB.IfthereflectedraymovesalongBC,findtheequationofBC.
Ans.:2x−y−6=0
55. Ifthelinesp1x+q1y=1,p2x+q2y=1,andp3x+q3y=1areconcurrent,provethatthepoints(p1,q1),(p2,q2),and(p3,q3)arecollinear.
56. Ifp,q,andrbethelengthoftheperpendicularsfromtheverticesA,B,andCofatriangleonany
straightline,provethata2(p–q)(p–r)+b2(q–r)(q–p)+c2(r–p)(r–q)=4Δ2.57. Provethattheareaoftheparallelogramformedbythestraightlinea1x+b1y+c1=0,a1x+b1y
+d1=0,a2x+b2y+c2=0,anda2x+b2y+d2=0is
58. Arayoflightissentalongtheline2x−3y=5.Afterrefractingacrossthelinex+y=1,itenterstheoppositesidesafterturningby15°awayfromthelinex+y=1.Findtheequationofthelinealongwhichtherefractedraytravels
Ans.:(15 −20)x−(30−10 )y+(11−18 )=0.
59. Twosidesofanisoscelestrianglearegivenbytheequations7x−y+3=0andx+y−7=0anditsthirdsidepassesthroughthepoint(1,−10).Determinetheequationofthethirdside.
Ans.:x−3y−31=0,3x+y+7=0.
60. Findallthosepointsonthelinex+y=4whichareatcunitdistancefromtheline4x+3y=10.61. Arethepoints(3,4)and(2,−6)onthesameoroppositesidesoftheline3x−4y=8?
Ans.:oppositesides
62. Howmanycirclescanbedrawneachtouchingallthethreelinesx+y=1,y=x,and7x−y=6?Findthecentreandradiusofoneofthecircles.
Ans.:Focus:(0,7)Incentre
63. Showthat beanypointonalinethentherangeofvaluesoftforwhichthe
pointpliesbetweentheparallellinesx+2y=1and2x+
64. Showthata,b,andcareanythreetermsofAPthenthelineax+by+c=0alwayspassesthroughafixedpoint.
65. Showthatifa,b,andcareinG.P.,thenthelineax+by+c=0formsatrianglewiththeaxes,whoseareaisaconstant.
Chapter3
PairofStraightLines
3.1INTRODUCTION
Weknowthateverylinearequationinxandyrepresentsastraightline.ThatisAx+By+C=0,whereA,BandCareconstants,representsastraightline.Considertwostraightlinesrepresentedbythefollowingequations:
Alsoconsidertheequation
If(x1,y1)isapointonthestraightlinegivenby(3.1)then
l1x1+m1y+n1=0Thisshowsthat(x1,y1)isalsoapointonthelocusof(3.3).Therefore,every
pointonthelinegivenby(3.1)isalsoapointonthelocusof(3.3).Similarly,everypointonthelinegivenby(3.2)isalsoapointonthelocusof(3.3).Therefore,(3.3)satisfiesallpointsonthestraightlinesgivenby(3.1)and(3.2).Hence,wesay(3.3)representsthecombinedequationofthestraightlinesgivenby(3.1)and(3.2).Itispossibletorewrite(3.3)as
Thepairofstraightlinesgivenby(3.1)and(3.2)isingeneralrepresentedintheform(3.4).However,wecannotsaythateveryequationofthisformwillrepresentapairofstraightlines.Wewillfindtheconditionthatisnecessaryandsufficientfortheequationoftheform(3.4)torepresentapairofstraightlines.Beforethatwewillseethateveryseconddegreehomogeneousequationinxandywillrepresentapairofstraightlines.
3.2HOMOGENEOUSEQUATIONOFSECONDDEGREEINxANDy
Everyhomogeneousequationofseconddegreeinxandyrepresentsapairofstraightlinespassingthroughtheorigin.Considertheequationax2+2hxy+by2=0,a≠0.
Dividingbyx2,weget Thisisaquadraticequationin
andhencetherearetwovaluesfor saym1andm2.Then
(i.e.)b(y−m1x)(y−m2x)=0.
Buty−m1x=0andy−m2x=0arestraightlinespassingthroughtheorigin.Therefore,ax2+2hxy+by2=0representsapairofstraightlinespassing
throughtheorigin.
Note3.2.1:ax2+2hxy+by2=b(y−m1x)(y−m2x)Equatingthecoefficientsofx2andxy,weget
3.3ANGLEBETWEENTHELINESREPRESENTEDBYax2+2hxy+by2=0
Lety–m1x=0andy−m2x=0bethetwolinesrepresentedbyax2+2hxy+by2
=0.
Letθbetheanglebetweenthelinesgivenbyax2+2hxy+by2=0.Thentheanglebetweenthelinesisgivenby
Thepositivesigngivestheacuteanglebetweenthelinesandthenegativesigngivestheobtuseanglebetweenthem.
Note3.3.1:Ifthelinesareparallelorcoincident,thenθ=0.Thentanθ=0.Therefore,from(3.7),wegeth2=ab.
Note3.1.3:Ifthelinesareperpendicularthen andsowegetfrom(3.7)
Thismeansa+b=0.Hence,theconditionforthelinestobe
parallelorcoincidentish2=abandtheconditionforthelinestobeperpendicularisa+b=0(i.e.)Coefficientofx2+Coefficientofy2=0.
3.4EQUATIONFORTHEBISECTOROFTHEANGLESBETWEENTHELINESGIVENBYax2+
2hxy+by2=0
Wewillnowderivetheequationforthebisectoroftheanglesbetweenthelinesgivenbyax2+2hxy+by2=0.Thecombinedequationofthebisectorsofthe
anglesbetweenthelinesgivenbyax2+2hxy+by2=0is
LetOAandOBbethetwolinesy−m1x=0andy−m2x=0representedbyax2
+2hxy+by2=0.LetthelinesOAandOBmakeanglesθ1andθ2withthex-axis.Then,weknowthat
LetθbetheanglemadebytheinternalbisectorOPwithOX.
Then istheanglemadebytheexternalbisectorOQwithOX.The
combinedequationofthebisectorsis
From(3.8)and(3.9),weget
Hence,thecombinedequationofthepairofbisectorsis
Aliter:Let(x1,y1)beapointonthebisectorOP.Then
Also,2θ=θ1+θ2.Accordingto(3.9)
From(3.9)and(3.10),
Thelocusof(x1,y1)is
Thisisthecombinedequationofthebisectors.
3.5CONDITIONFORGENERALEQUATIONOFASECONDDEGREEEQUATIONTOREPRESENTAPAIROFSTRAIGHTLINES
Wewillnowderivetheconditionforthegeneralequationofaseconddegreeequationtorepresentapairofstraightlines.Theconditionforthegeneralequationoftheseconddegreeax2+2hxy+by2+2gx+2fy+c=0torepresentapairofstraightlinesisabc+2fgh−af2−bg2−ch2=0.
Method1:Considerthegeneralequationoftheseconddegree
Letlx+my+n=0andl1x+m1y+n1=0betheequationsoftwolinesrepresentedby(3.11).Then
Comparingthecoefficients,weget
Weknowthat
Bymultiplyingthetwodeterminants,weget
Bymultiplyingthetwodeterminants,weget
Substitutingthevaluesfrom(3.12)in(3.13),weget
Expandingthedeterminant,weget
Thisistherequiredcondition.
Method2:
ax2+2hxy+by2+2gx+2fy+c=0Writingthisequationintheformby2+2hxy+2fy+(ax2+2gx+c)=0andsolvingforyweget
Thisequationwillrepresenttwostraightlinesifthequadraticexpressionundertheradicalsignisaperfectsquare.Theconditionforthisis4(hf−bg)2−4(h2−ab)(f2−bc)=0
Sinceb≠0,
abc+2fgh−af2−bg2−ch2=0Thisistherequiredcondition.
Method3:Lettheequationax2+2hxy+by2+2gx+2fy+c=0representapairofstraightlinesandlet(x1,y1)betheirpointofintersection.Shiftingtheorigintothepoint(x1,y1),weget
wherethenewaxesOXandOYareparallelto(Ox,Oy).
As(3.16)representsapairofstraightlinespassingthroughtheneworigin,ithastobeahomogeneousequationinXandY.Hence,
Substituting(3.17)and(3.18)in(3.20),weget
gx1+fy1+c=0(3.21)Eliminatingx1andy1from(3.17),(3.18)and(3.21),weget
Expanding,wegetabc+2fgh−af2−bg2−ch2=0.
Note3.5.1:Solving(3.17)and(3.18),weget
Hence,thepointofintersectionofthelinesrepresentedby(3.11)is
Note3.5.2:Iflx+my+n=0andl1x+m1y+n1=0arethetwostraightlinesrepresentedby(3.11),thenlx+my=0andl1x+m1y=0willrepresenttwostraightlinesparalleltothelinesrepresentedby(3.11)andpassingthroughtheorigin.Theircombinedequationis
Therefore,ifax2+2hxy+by2+2gx+2fy+c=0representsapairofstraightlines,thentheequationax2+2hxy+by2=0willrepresentapairoflinesparalleltothelinesgivenby(3.11).Weknowthateveryhomogeneousequationofseconddegreeinxandy
representsapairofstraightlinespassingthroughtheorigin.Wenowusethisideatogetthecombinedequationofthepairoflinesjoiningtheorigintothepointofintersectionofthecurveax2+2hxy+by2+2gx+2fy+c=0andthelinelx+my=1.
Theequationofthecurve
andtheline
lx+my=1.(3.23)willmeetattwopointssayPandQ.Let(x1,y1)beoneofthepointsofintersection,sayP.Then
andlx1+my1=1Letushomogenise(3.22)withthehelpof(3.23).Then,wewrite
Ifwesubstitutex=x1andy=y1in(3.25),weget
becauseof(3.23)and(3.24).ThereforeP(x1,y1)liesonthelocusof(3.25).SimilarlywecanshowthatthepointQ(x2,y2)alsoliesonthelocusof(3.25).However,aseconddegreehomogeneousequationrepresentsapairofstraightlinespassingthroughorigin.Hence,(3.25)isthecombinedequationofthepairoflinesOPandOQ.Hence,homogensingtheseconddegreeequation(3.22)withthehelpof
(3.23),wegetapairofstraightlinespassingthroughtheorigin.
ILLUSTRATIVEEXAMPLES
Example3.1
Thegradientofoneofthelinesax2+2hxy+by2=0istwicethatoftheother.Showthat8h2=9ab.
Solution
Theequationax2+2hxy+by2=0representsapairofstraightlinespassingthroughtheorigin.Letthelinesbey−m1x=0andy−m2x=0.Then
ax2+2hxy+by2=b(y–m1x)(y=m2x)Equatingthecoefficientsofxyandx2onbothsides,weget
Here,ithasbeengiventhatm2=2m1.
From(3.26)and(3.27),weget
Example3.2
Provethatoneofthelinesax2+2hxy+by2=0willbisectananglebetweenthecoordinateaxesif(a+b)2=4h2.
Solution
Lety–m1x=0andy–m2x=0bethetwolinesrepresentedbya2+2hxy+by2
=0.
Then
Sinceoneofthelinesbisectstheanglebetweentheaxes,wetakem1=±1.Then
Example3.3
Findthecentroidofthetriangleformedbythelinesgivenbytheequations12x2
–20xy+7y2=0and2x–3y+4=0.
Solution
Therefore,thesidesofthetrianglearerepresentedby
Thepointofintersectionofthelines(3.28)and(3.29)is(0,0).Letussolve(3.29)and(3.30).
Thus,thepointofintersectionofthesetwolinesis(7,6).Now,letussolve(3.28)and(3.30).
Thus,thepointofintersectionofthesetwolinesis(1,2).Then,thecentroidof
thetrianglewithvertices(0,0),(7,6)and(1,2)is (i.e)
Example3.4
Findtheproductofperpendicularsdrawnfromthepoint(x1,y1)onthelinesax2
+2hxy+by2=0.
Solution
Letthelinesbey–m1x=0andy−m2x=0.Then
Letp1andp2betheperpendiculardistancesfrom(x1,y1)onthetwolinesy–m1x=0andy–m2x=0,respectively.
Example3.5
Ifthelinesax2+2hxy+by2=0bethetwosidesofaparallelogramandthelinelx+my=1beoneofthediagonals,showthattheequationoftheotherdiagonalisy(bl–hm)y=(am–h)lx.Showthattheparallelogramisarhombusifh(a2–b2)=(a–h)lm.
Solution
ThediagonalACnotpassingthroughtheoriginislx+my=1.
TheequationofthelinesOAandOCbey−m1x=0andy−m2x=0.ThenthecorrespondingcoordinatesofAaregotbysolvingy–m1x=0andlx
+my=1.
Sincediagonalsbisecteachotherinaparallelogram,theequationofthediagonal
OBis
IfOABCisarhombus,thenthediagonalsareatrightanglesHence,theproductoftheirslopesis–1.
Example3.6
Provethattheareaofthetriangleformedbythelinesy=x+candthestraight
lines
Solution
Letthetwolinesrepresentedbyax2+2hxy+by2=0bey–m1x=0andy–m2x=0.Solvingtheequationsy–m1x=0andy=x+c,wegetthecoordinatesofAtobe
Example3.7
LandMarethefeetoftheperpendicularsfrom(c,0)onthelinesax2+2hxy+by2=0.ShowthattheequationofthelineLMis(a–b)x+2hy+bc=0.
Solution
LettheequationofLMbelx+my=1.SinceLandMarethefeetoftheperpendicularsfromA(c,0)onthetwolines
y−m1x=0andy–m2x=0,thepointsO,A,LandMareconcyclic.TheequationofthecirclewithOAasdiameterisx(x–c)+y2=0orx2+y2–cx=0.ThecombinedequationofthelinesOLandOMisgotbyhomogenisingthe
equationofthecirclewiththehelpoflinelx+my=1.Hence,thecombinedequationofthelinesOLandLMis
ButthecombinedequationofthelinesOLandOMis
ax2+2hxy+by2=0Boththeseequationsrepresentthesamelines.Thereforeidentifyingtheseequations,weget
Therefore,thelinelx+my–1=0is
(i.e.)(a–b)x+2hy+bc=0
Example3.8
Showthatfordifferentvaluesofpthecentroidofthetriangleformedbythestraightlinesax2+2hxy+by2=0arexcosα+ysinα=pliesonthelinex(atanα–h)+y(htanα–b)=0.
Solution
LetOAandOBbethelinesrepresentedbyax2+2hxy+by2=0andtheirequationsbey–m1x=0andy–m2x=0.TheequationofthelineABisxcosα+ysinα=p.ThecoordinatesofAare
ThecoordinatesofBare
Themidpoint(x1,y1)ofABis
Example3.9
Findtheconditionthatoneofthelinesgivenbyax2+2hxy+by2=0maybeperpendiculartooneofthelinesgivenbya1x2+2h1xy+by2=0.
Solution
Lety=mxbealineofax2+2hxy+by2=0.Then
Then
Hence,
From(3.31)and(3.32),weget
Hence,therequiredconditionis(aa1–bb1)2+4(ha1+h1b)(bh1+a1h)=0.
Example3.10
Twosidesofatriangleliealongy2–m2x2=0anditsorthocentreis(c,d).Showthattheequationofitsthirdsideis(1–m2)(cx+dy)=c2–m2d2.
Solution
LetOA,OBandABbethelines
EquationofODisbx–ay=0.ThispassesthroughH(c,d).∴bc=ad.(1)EquationofAHis
ThecoordinatesofAare
Thatpointliesx–my=c–md
From(3.33),
Hence,theequationofthelineAB(ax+by=1)becomes(1–m2)(cx+dy)=c2
–m2d2.
Example3.11
Showthattheequationm(x3−3xy2)+y3–3x2y=0representsthreestraightlinesequallyinclinedtooneanother.
Solution
y3–3x2y=m(3xy2–x3)Dividingbyx3,weget
Thesevaluesofθshowthatthelinesareequallyinclinedtooneanother.
Example3.12
Showthatthestraightlines(A2–3B2)x2+8ABx+(B2–3A2)=0formwiththe
lineAx+By+C=0anequilateraltriangleofarea
Solution
Thesidesofthetrianglearegivenby
Theanglebetweenthelines(3.34)and(3.36)is
Similarlytheanglebetweenthelines(3.35)and(3.36)is
Sincethethreesidesformatriangle, istheonlypossibility.
Hence,thetriangleisequilateral.
Example3.13
Showthattwoofthestraightlinesax3+bx2y+cxy2+dy3=0willbeperpendiculartoeachotherifa2+d2+bd+ac=0.
Solution
ax3+bx2y+cxy2+dy3=0Thisbeingathirddegreehomogeneousequation,itrepresentsthreestraightlinespassingthroughorigin.Letthethreelinesbey–m1x=0,y–m2x=0andy–m3x=0.Ifmistheslopeofanylinethen
From(3.38)and(3.39),weget
Sincem3isarootof(3.37)
Exercises
1. Showthattheequationofpairoflinesthroughtheoriginandperpendiculartothepairoflinesax2
+2hxy+by2=0isbx2–2hxy+ay2=0.2. ThroughapointAonthex-axis,astraightlineisdrawnparalleltothey-axissoastomeetthepair
ofstraightlinesax2+2hxy+by2=0inBandC.IfAB=BC,provethat8h2=9ab.3. FromapointA(1,1),straightlinesALandAMaredrawnatrightanglestothepairofstraightlines
3x2+7xy–2y2=0.FindtheequationofthepairoflinesALandAM.AlsofindtheareaofthequadrilateralALOMwhereOistheoriginofthecoordinate.
4. Showthattheareaofthetriangleformedbythelinesax2+2hxy+by2=0andlx+my=1is
5. Showthattheorthocentreofthetriangleformedbythelinesax2+2hxy+by2=0andlx+my=1
isgivenby
6. Showthatthecentroid(x1,y1)ofthetriangleformedbythelinesax2+2hxy+by2=0andlx+
my=1is
7. Atrianglehasthelinesax2+2hxy+by2=0fortwoofitssidesandthepoint(c,d)forits
orthocentre.Provethattheequationofthethirdsideis(a+b)(cx+dy)=ad2–2hbd+bc2.
8. Iftheslopeofoneofthelinesgivenbyax2+2hxy+by2=0isktimestheother,provethat4kh2
=abc(1+k)2.9. Ifthedistanceofthepoint(x1,y1)fromeachoftwostraightlinesthroughtheoriginisd,prove
thattheequationofthestraightlinesis(x1y–xy1)2=d2(x2+y2).
10. Astraightlineofconstantlength2lhasitsextremitiesoneoneachofthestraightlinesax2+2hxy
+by2=0.Showthatthelineofmidpointis(ax+by)2(hx+by)+(ab–h2)2l2a.
11. Provethatthetriangleformedbythelinesax2+2hxy+by2=0andlx+my=1isrightangledif
(a+b)/al2+2hlm+bm2)=0.
12. Showthatiftwoofthelinesax3+bx2y+cxy2+dy3=0makecomplementaryangleswithx-axisinanticlockwisedirection,thena(a–c)+d(b–d)=0.
13. Iftheslopeofthelinesgivenbyax2+2hxy+by2=0isthesquareoftheother,showthatab(a+
h)–6ahb+8h3=0.
14. Showthatthelineax+by+c=0andthetwolinesgivenby(ax+by)2=3(bx–ay)2forman
equilateraltriangleofarea
15. Ifoneofthelinegivenbyax2+2hxy+by2=0iscommonwithoneofthelinesofa1x2+2h1xy
+b1y2=0.showthat(ab1–a1b)
2+4(ah1–a1h).(bh1–b1h)=0.
16. Apointmovessothatitsdistancebetweenthefeetoftheperpendicularsfromitonthelinesax2+
2hxy+by2=0isaconstant2k.Showthatthelocusofthepointis(x2+y2)(h2–ab)=k2[(a–b)2
+4h2].17. Showthatthedistancefromtheorigintotheorthocentreofthetriangleformedbythelines
andax2+2hxy+by2=0is
18. Aparallelogramisformedbythelinesax2+2hxy+by2=0andthelinesthrough(p,q)paralleltothem.Showthattheequationofthediagonalnotpassingthroughtheoriginis(2x–p)(ap+hq)+(2y–q)(hp+bq)=0.
19. Ifthelinesgivenbylx+my=1andax2+2hxy+by2=0formanisoscelestriangle,showthat
h(l2–m2)=lm(a–b).
Example3.14
Findλsothattheequationx2+5xy+4y2+3x+2y+λ=0representsapairoflines.Findalsotheirpointofintersectionandtheanglebetweenthem.
Solution
Considertheseconddegreetermsx2+5xy+4y2.
x2+5xy+4y2=(x+y)(x+4y)Letthetwostraightlinesbex+y+l=0andx+4y+m=0.Then
Equatingthecoefficientsofx,yandconstantterms,weget
Solving(3.40)and(3.41),weget
From(3.42),
Thenthetwolinesare and3x+12y+
10=0.
Theanglebetweenthelinesisgivenby
Example3.15
Findthevalueofλsothattheequationλx2–10xy+12y2+5x–16y–3=0representsapairofstraightlines.Findalsotheirpointofintersection.
Solution
λx2–10xy+12y2+5x–16y–3=0Comparingwiththeequationax2+2hxy+by2+2gx+2fy+c=0wegeta=λ,2h=−10,b=12,2g=5,2f=–16,c=–3
Theconditionforthegivenequationtorepresentapairofstraightlinesisabc+2fgh–af2–bg2–ch2=0
–36λ+200–64λ–75+75=0⇒λ=2Then2x2–10xy+12y2+5x–16y–3=(2x–4y+l)(x−3y+m)Equatingthecoefficientsofx,yandconstantterms,
Therefore,thetwolinesarex–2y+3=0and2x−6y−1=0.Solvingthesetwo
equations,wegetthepointofintersectionas
Example3.16
Findthevalueofλsothattheequationx2−λxy+2y2+3x−5y+2=0representsapairofstraightlines.
Solution
Example3.17
Provethatthegeneralequationoftheseconddegreeax2+2hxy+by2+2gx+2fy+c=0representsparallelstraightlinesifh2=abandbg2=af2.Provethat
thedistancebetweenthetwostraightlinesis
Solution
Lettheparallellinesbelx+my+n=0andlx+my+n1=0.Thenax2+2hxy+by2+2gx+2fy+c=(lx+my+n)(lx+my+n1)
Equatingtheliketerms,weget
Also,thedistancebetweenthelineslx+my+n=0andlx+my+n1=0is
Example3.18
Ifax2+2hxy+by2+2gx+2fy+c=0representstwostraightlinesequidistantfromtheorigin,showthatf4−g4=c(bf2−ag2).
Solution
Letthetwolinesrepresentedbythegivenequationbelx+my+n=0andl1x+m1y+n1=0.Then
Perpendiculardistancesfromtheorigintothetwolinesareequal.Therefore,
Squaring
Example3.19
Iftheequationax2+2hxy+by2+2gx+by2+2gx+2fy+c=0representstwostraightlines,provethattheproductofthelengthsoftheperpendicularsfromthe
originonthestraightlinesis
Solution
Letthetwolinesbelx+my+n=0andl1x+my+n=0.Therefore
Theproductoftheperpendicularsfromtheoriginontheselines
Example3.20
Ifax2+2hxy+by2+2gx+by2+2gx+2fy+c=0representstwostraightlines,provethatthesquareofthedistanceoftheirpointofintersectionfromtheorigin
is Further,ifthetwogivenlinesareperpendicular,thenprove
thatthedistanceoftheirpointofintersectionfromtheoriginis
Solution
Letthetwostraightlinesbelx+my+n=0andl1x+m1y+n1=0.
Theirpointofintersectionis
Hence,thedistanceofthispointfromtheoriginisgivenby
Ifthelinesareperpendicularthen(a+b)=0.Then
Example3.21
Showthatthelinesgivenby12x2+7xy−12y2=0and12x2+7xy−12y2−x+7y−1=0arealongthesidesofasquare.
Solution
Theseconddegreetermsin(3.42)and(3.43)arethesame.Thisimpliesthatthetwolinesrepresentedby(3.42)areparalleltothetwolinesrepresentedby(3.43).Hence,thesefourlinesfromaparallelogram.Also,ineachoftheequationscoefficientofx2+coefficientofy2=0.Hence,eachequationformsapairofperpendicularlines.Thus,thefourlines
formarectangle.Thetwolinesrepresentedby(3.42)are3x+4y=0and4x−3y=0.Thetwolinesrepresentedby(3.43)are3x+4y−1=0and4x−3y+1=0.
Theperpendiculardistancebetween2x+4y=0and3x+4y−1=0is .
Theperpendiculardistancebetween4x−3y=0and4x−3y+1=0is .
Hence,thefourlinesformasquare.
Exercises
1. Showthattheequation6x2+17xy+12y2+22x+31y+20=0representsapairofstraightlinesandfindtheirequations.
Ans.:2x+3y+4=03x+4y+5=0
2. Provethattheequations8x2+8xy+2y2+26x+13y+15=0representstwoparallelstraightlinesandfindthedistancebetweenthem.
Ans.:
3. Provethattheequation3x2+8xy−7y2+21x−3y+18=0representstwolines.Findtheirpointofintersectionandtheanglebetweenthem.
Ans.:
4. Ifax2+2hxy+by2+2gx+2fy+c=0andax2+2hxy+by2−2gx−2fy+c=0eachrepresentsa
pairoflines,provethattheareaoftheparallelogramenclosedis
5. Showthattheequation3x2+10xy+8y2+14x−22y+15=0representstwostraightlines
intersectingatanangle
6. Theequationax2−2xy−2y2−5x+5y+c=0representstwostraightlinesperpendiculartoeachother.Findaandc.
Ans.:a=2,c=−3
7. Findthedistancebetweentheparallellinesgivenby4x2+12xy+9y2−6x−9y+1=0.
Ans.:
8. Showthatthefourlines2x2+3xy−2y2=0and2x2+3xy−2y2−3x+y+1=0formasquare.
9. Showthatthestraightlinesrepresentedbyax2+2hxy+by2=0andthoserepresentedbyax2+
2hxy+by2+2gx+2fy+c=0formarhombus,if(c−h)fg+h(f2−g2)=0.
10. Iftheequationax2+2hxy+by2+2gx+2fy+c=0representstwostraightlinesandparallellinestothesetwolinesaredrawnthroughtheoriginthenshowthattheareaoftheparallelogramso
formedis
11. Ifthestraightlinesgivenbyax2+2hxy+by2+2gx+2fy+c=0intersectsonthey-axisthen
showthat2fgh−hg2−ch2=0.
12. Aparallelogramissuchthattwoofitsadjacentsidesarealongthelinesax2+2hxy+by2=0anditscentreis(a,b).Findtheequationoftheothertwosides.
Ans.:a(x−2a)+2h(x−2a)(y−2b)+b(y−2b)2=0
Example3.22
Showthatthepairoflinesgivenby(a−b)(x2−y2)+4hxy=0andthepairoflinesgivenbyh(x2−y2)=(a−b)xyaresuchthateachpairbisectstheanglebetweentheotherpairs.
Solution
Thecombinedequationofthebisectorsofthepairoflinesgivenby(3.44)is
(i.e.)h(x2−y2)=xy(a−b)
whichis(3.45).Thecombinedequationofthebisectorsoftheanglebetweenlinesgivenby
(3.45)is
whichis(3.44).Hence,eachpairbisectstheanglebetweentheother.
Example3.23
Ifthebisectorsofthelinex2−2pxy−y2=0arex2−2qxy−y2=0showthatpq+1=0.
Solution
Thecombinedequationofthebisectorsof(3.46)is
Butequationofthebisectorisgivenby
x2−2qxy−y2=0(3.47)
Comparing(3.46)and(3.47),weget
∴pq+1=0
Example3.24
Provethatifoneofthelinesgivenbytheequationax2+2hxy+by2=0bisectstheanglebetweenthecoordinateaxesthen(a+b)2=4h2.
Solution
Thebisectorsofthecoordinateaxesaregivenbyy=xandy=−x.Ify=xisoneofthelinesofax2+2hxy+by2=0thenax2+2hx2+bx2=0.
(i.e.)a+b=–2hIfy=–xisoneofthelinesofax2+2hxy+by2=0,thena+b=2h.Fromthesetwoequations,weget(a+b)2=4h2.
Example3.25
Showthattheliney=mxbisectstheanglebetweenthelinesax2+2hxy+by2=0ifh(1−m2)+m(a−b)=0.
Solution
Thecombinedequationofthebisectorsoftheanglesbetweenthelinesax2−2hxy+by2=0is
Ify=mxisoneofthebisectors,thenithastosatisfytheaboveequation.
Example3.26
Showthatthepairofthelinesgivenbya2x2+2h(a+b)xy+b2y2=0isequallyinclinedtothepairgivenbyax2+2hxy+by2=0.
Solution
Inordertoshowthatthepairoflinesgivenbya2x2+2h(a+b)xy+b2y2=0isequallyinclinedtothepairoflinesgivenbyax2+2hxy+by2=0,wehavetoshowthatboththepairshavethesamebisectors.Thecombinedequationsofthe
bisectorsofthefirstpairoflinesis whichisthe
combinedequationofthesecondpairoflines.
Exercises
1. Ifthepairoflinesx2−2axy−y2=0bisectstheanglesbetweenthelinesx2−2pxy−y2=0thenshowthatthelatterpairalsobisectstheanglebetweentheformerpair.
2. Ifoneofthebisectorsofax2+2hxy+by2=0passesthroughthepointofintersectionofthelines
ax2+2hxy+by2+2gx+2fy+c=0thenshowthath(f2−g2)+(a−b)fg=0.
3. Ifthepairofstraightlinesax2+2hxy+by2=0andbx2+2gxy+by2=0besuchthateachbisectstheanglebetweentheotherthenprovethathg−b=0.
4. Provethattheequations6x2+xy−12y2−14x+47y−40=0and14x2+xy−4y2−30x+15y=0representtwopairsoflinessuchthatthelinesofthefirstpairareequallyinclinedtothoseofthesecondpair.
5. Provethattwoofthelinesrepresentedbytheequationax4+bx2y+cx2y2+dxy3+ay4=0willbisecttheanglebetweentheothertwoifc+ba=0andb+d=0.
Example3.27
Ifthestraightlinesjoiningtheorigintothepointofintersectionof3x2−xy+3y2
+2x−3y+4=0and2x+3y=kareatrightangles,provethat6k2−5k+52=0.
Solution
Let
Thecombinedequationofthelinesjoiningtheorigintothepointofintersectionofthelinesgiven(3.48)and(3.49)isgotbyhomogenising(3.48)withthehelpof(3.49).Hence,thecombinedequationofthelinesjoiningtheorigintothepointsofintersectionof(3.48)and(3.49)is
Sincethetwostraightlinesareatrightangles,coefficientofx2+coefficientofy2=0
Example3.28
Showthatthepairofstraightlinesjoiningtheorigintothepointofintersectionofthestraightlinesy=mx+candthecirclex2+y2=a2areatrightangles2c2
=a2(1+m2).
Solution
Itisgiventhatx2+y2=a2andy=mx+c.
ThecombinedequationofthelinesOPandOQisgivenby
SinceOPandOQareatrightangles,coefficientofx2+coefficientofy2=0
c2−m2a2+c2−a2=0⇒2c2=a2(1+m2)
Example3.29
Showthatthejoinoforigintotheintersectionofthelines2x2−7xy+3y2+5x+10y−25=0andthepointsatwhichtheselinesarecutbythelinex+2y−5=0aretheverticesofaparallelogram.
Solution
Letequation(3.50)representsthelinesCAandCBand(3.51)representsthelineAB.ThecombinedequationofthelinesOAandOBisgotbyhomogeniousing
(3.50)withthehelpof(3.51).
Sincetheseconddegreetermsin(3.50)and(3.52)arethesamethetwolinesrepresentedby(3.50)areparalleltothetwolinesrepresentedby(3.52).Therefore,thefourlinesformaparallelogram.
Example3.30
Ifthechordofthecirclex2+y2=a2whoseequationislx+my=1subtendsanangleof45°attheoriginthenshowthat4[a2(l2+m2)−1]=[a2(l2+m2)−2]2.
Solution
Itisgiventhat,
ThecombinedequationofthelinesOPandOQis
Then
Example3.31
Findtheequationtothestraightlinesjoiningtheorigintothepointof
intersectionofthestraightline andthecircle5(x2+y2+ax+by)=9ab
andfindtheconditionsthatthestraightlinesmaybeatrightangles.
Solution
Itisgiventhat,
Thecombinedequationofthelinesjoiningtheorigintothepointsofintersectionof(3.53)and(3.54)is
Sincethelinesareatrightangles,coefficientofx2+coefficientofy2=0
Example3.32
Thelinelx+my=1meetsthecirclex2+y2=a2inPandQ.IfOistheorigin
thenshowthat .
Solution
Theperpendicularfromtheorigintotheline OP=a
Example3.33
Thestraightliney−k=m(x+2a)intersectsthecurvey2=4a(x+a)inAandC.
Showthatthebisectorsofangle ,‘O’beingtheorigin,arethesameforall
valuesofm.
Solution
Let
ThecombinedequationofthelinesOAandOBis
Thecombinedequationofthebisectorsis
Example3.34
Provethatifallchordsofax2+2hxy+by2+2gx+2fy+c=0subtendarightangleattheorigin,thentheequationmustrepresenttwostraightlinesatrightanglesthroughtheorigin.
Solution
Lettheequationofthechordbe
lx+my=1(3.56)Letthelines(3.55)and(3.56)intersectatPandQ.ThecombinedequationofOPandOQisax2+2hxy+by2+(2gx+2fy)(lx+my)+c(lx+my)2=0.
Since ,coefficientofx2+coefficientofy2=0.
(a+2gl+cl2)+(b+2fm+cm2)=0
Sincelandmarearbitrary,coefficientsofl1,l2,m1,m2andtheconstanttermvanishseparately.Sinceg=0,f=0,c=0anda+b=0.Hence,equation(3.55)becomesax2+2hxy+by2=0whichisapairof
perpendicularlinesthroughtheorigin.
Exercises
1. Showthatthelinejoiningtheorigintothepointscommonto3x2+5xy+3y2+2x+3y=0and3x−2y=1areatrightangles.
2. Ifthestraightlinesjoiningtheorigintothepointofintersection3x2−xy+3y2+2x−3y+4=0
and2x+3y=kareatrightanglesthenshowthat6k2−5k+52=0.
3. Showthatallthechordsofthecurve3x2−y2−2x+y=0whichsubtendarightangleattheoriginpassthroughafixedpoint.
4. Ifthecurvex2+y2+2gx+2fy+c=0interceptsonthelinelx+my=1,whichsubtendsaright
angleattheoriginthenshowthata(l2+m2)+2(gl+fm+1)=0.5. Ifthestraightlinesjoiningtheorigintothepointofintersectionofthelinekx+hy=2hkwiththe
curve(x−h)2+(y−k)2=a2areatrightanglesattheoriginshowthath2+k2=a2.
6. Provethatthetriangleformedbythelinesax2+2hxy+by2=0andlx+my=1isisoscelesif(l2
−m2)h=(a−b)lm.
7. Provethatthepairoflinesjoiningtheorigintotheintersectionofthecurves bythe
linelx+my+n=0arecoincidentifa2l2+b2m2=n2.
8. Showthatthestraightlinesjoiningtheorigintothepointofintersectionofthecurvesax2+2hxy
+by2+2gx=0anda1x2+2h1xy+b1y
2+2g1x=0willbeatrightanglesifg1(a1+b1)=g(h1+b1).
9. Showthattheanglebetweenthelinesdrawnfromtheorigintothepointofintersectionofx2+2xy
+y2+2x+2y−5=0and3x−y+1=0is
Chapter4
Circle
4.1INTRODUCTION
Definition4.1.1:Acircleisthelocusofapointinaplanesuchthatitsdistancefromafixedpointintheplaneisaconstant.Thefixedpointiscalledthecentreofthecircleandtheconstantdistanceiscalledtheradiusofthecircle.
4.2EQUATIONOFACIRCLEWHOSECENTREIS(h,k)ANDRADIUSr
LetC(h,k)bethecentreofthecircleandP(x,y)beanypointonthecircle.CP=ristheradiusofthecircle.CP2=r2(i.e.)(x−h)2+(y−k)2=r2.Thisistheequationoftherequiredcircle.
Note4.2.1:Ifthecentreofthecircleisattheorigin,thentheequationofthecircleisx2+y2=r2.
4.3CENTREANDRADIUSOFACIRCLEREPRESENTEDBYTHEEQUATIONx2+y2+2gx+2fy+c=0
Addingg2+f2tobothsides,weget
Thisequationisoftheform(x−h)2+(y−k)2=r2,whichisacirclewithcentre(h,k)andradiusr.Thus,equation(4.1)representsacirclewhosecentreis(−g,
−f)andradius
Note4.3.1:Aseconddegreeequationinxandywillrepresentacircleifthecoefficientsofx2andy2areequalandthexytermisabsent.
Note4.3.2:
1. Ifg2+f2−cispositive,thentheequationrepresentsarealcircle.
2. Ifg2+f2−ciszero,thentheequationrepresentsapoint.
3. Ifg2+f2−cisnegative,thentheequationrepresentsanimaginarycircle.
4.4LENGTHOFTANGENTFROMPOINTP(x1,y1)TOTHECIRCLEx2+y2+2gx+2fy+c=0
ThecentreofthecircleisC(−g,−f)andradius .LetPTbethe
tangentfromPtothecircle.
Note4.4.1:
1. IfPT2>0thenpointP(x1,y1)liesoutsidethecircle.
2. IfPT2=0thenthepointP(x1,y1)liesonthecircle.
3. IfPT2<0thenpointP(x1,y1)liesinsidethecircle.
4.5EQUATIONOFTANGENTAT(x1,y1)TOTHECIRCLEx2+y2+2gx+2fy+c=0
Thecentreofthecircleis(−g,−f).Theslopeoftheradius
Hence,theequationoftangentat(x1,y1)is(y−y1)=m(x−x1)
Addinggx1+fy1+ctobothsides,
sincethepoint(x1,y1)liesonthecircle.Hence,theequationofthetangentat(x1,y1)isxx1+yy1+g(x+x1)+f(y+y1)
+c=0.
4.6EQUATIONOFCIRCLEWITHTHELINEJOININGPOINTSA(x1,y1)ANDB(x2,y2)ASTHEENDSOFDIAMETER
A(x1,y1)andB(x2,y2)aretheendsofadiameter.LetP(x,y)beanypointonthe
circumferenceofthecircle.Then (i.e.)AP⊥PB.
TheslopeofAPis theslopeofBPis
SinceAPisperpendiculartoPB,m1m2=−1
Thisistherequiredequationofthecircle.
4.7CONDITIONFORTHESTRAIGHTLINEy=mx+cTOBEATANGENTTOTHECIRCLEx2+
y2=a2
Method1:Thecentreofthecircleis(0,0).Theradiusofthecircleisa.Ify=mx+cisatangenttothecircle,theperpendiculardistancefromthecentreonthestraightliney=mx+cistheradiusofthecircle.
Thisistherequiredcondition.
Method2:Theequationofthecircleis
x2+y2=a2(4.2)Theequationofthelineis
y=mx+c(4.3)Thex-coordinatesofthepointofintersectionofcircle(4.2)andline(4.3)aregivenby
Ify=mx+cisatangenttothecircle,thenthetwovaluesofxgivenbyequation(4.4)areequal.Theconditionforthisisthediscriminantofquadraticequation(4.4)iszero.
Thisistherequiredcondition.
Note4.7.1:Anytangenttothecirclex2+y2=a2isoftheform
4.8EQUATIONOFTHECHORDOFCONTACTOFTANGENTSFROM(x1,y1)TOTHECIRCLE
x2+y2+2gx+2fy+c=0
LetQRbethechordofcontactoftangentsfromP(x1,y1).LetQandRbethepoints(x2,y2)and(x3,y3),respectively.TheequationsoftangentsatQandRare
xx2+yy2+g(x+x2)+f(y+y2)+c=0xx3+yy3+g(x+x3)+f(y+y3)+c=0
ThesetwotangentspassthroughthepointP(x1,y1).Therefore,x1x2+y1y2+g(x1+x2)+f(y1+y2)+c=0and
x1x3+y1y3+g(x+x3)+f(y+y3)+c=0
Thesetwoequationsshowthatthepoints(x2,y2)and(x3,y3)lieonthestraightline
xx1+yy1+g(x+x1)+f(y+y1)+c=0
Hence,theequationofthechordofcontactfrom(x1,y1)is
xx1+yy1+g(x+x1)+f(y+y1)+c=0
4.9TWOTANGENTSCANALWAYSBEDRAWNFROMAGIVENPOINTTOACIRCLEAND
THELOCUSOFTHEPOINTOFINTERSECTIONOFPERPENDICULARTANGENTSISACIRCLE
Lettheequationofthecirclebe
x2+y2=a2(4.6)Let(x1,y1)beagivenpoint.Anytangenttothecirclex2+y2=a2is
Ifthistangentpointsthrough(x1,y1),then
Thisisaquadraticequationinm.Hence,therearetwovaluesform,andforeachvalueofmthereisatangent.Thus,therearetwotangentsfromagivenpointtoacircle.Let(x1,y1)bethepointofintersectionofthetwotangentsfrom(x1,y1).Ifm1andm2aretheslopesofthetwotangents,then
Ifthetwotangentsareperpendicular,thenm1m2=−1.
Thelocusof(x1,y1)isx2+y2=a2+b2,whichisacircle.
4.10POLEANDPOLAR
Definition4.10.1:Thepolarofapointwithrespecttoacircleisdefinedtobethelocusofthepointofintersectionoftangentsattheextremitiesofavariablechordthroughthatpoint.Thepointiscalledthepole.
4.10.1PolarofthePointP(x1,y1)withRespecttotheCirclex2+y2+2gx+2fy
+c=0
Lettheequationofcirclebe
LetQRbeavariablechordthroughthepointP(x1,y1).LetthetangentsatQandRtothecircleintersectatT(h,k).Then,QRisthechordofcontactofthetangentsfromT(h,k).Itsequationis
xh+yk+g(x+h)+f(y+k)+c=0
ThischordpassesthroughP(x1,y1).Therefore,
Thelocusof(h,k)is
Hence,thepolarof(x1,y1)isxx1+yy1+g(x+x1)+f(y+y1)+c=0.
Note4.10.1.1:
1. Ifthepoint(x1,y1)liesoutsidethecircle,thepolarof(x1,y1)isthesameasthechordofcontactfrom(x1,y1).Ifthepointliesonthecircle,thenthetangentat(x1,y1)isthepolarofthepointP(x1,y1).
2. Thepoint(x1,y1)iscalledthepoleofthelinexx1+yy1+g(x+x1)+f(y+y1)+c=0.Line(4.12)iscalledthepolarofthepoint(x1,y1).
3. Thepolarof(x1,y1)withrespecttothecirclex2+y2=a2isxx1+yy1=a
2.
4.10.2PoleoftheLinelx+my+n=0withRespecttotheCirclex2+y2=a2
Let(x1,y1)bethepoleoftheline
lx+my+n=0(4.13)withrespecttothecirclex2+y2=a2.Then,thepolarof(x,y)is
xx1+yy1=a2(4.14)Equations(4.13)and(4.14)representthesameline.Therefore,identifyingthesetwoequations,weget
Hence,thepoleofthelinelx+my+n=0is
4.11CONJUGATELINES
Definition4.11.1:Twolinesaresaidtobeconjugatewithrespecttothecirclex2
+y2=a2ifthepoleofeitherlineliesontheotherline.
4.11.1ConditionfortheLineslx+my+n=0andl1x+m1y+n1=0tobeConjugateLineswithRespecttotheCirclex2+y2=a2
Thepoleofthelinelx+my+n=0is Sincethetwogivenlinesare
conjugatetoeachother,thispointliesonthelinel1x+m1y+n1=0.
4.12EQUATIONOFACHORDOFCIRCLEx2+y2+2gx+2fy+c=0INTERMSOFITSMIDDLEPOINT
LetPQbeachordofthecirclex2+y2+2gx+2fy+c=0andR(x1,y1)beitsmiddlepoint.Theequationofanychordthrough(x1,y1)is
Anypointonthislineisx=x1+rcosθ,y=y1+rsinθ.WhenthechordPQmeetsthecirclethispointliesonthecircle.Therefore,
ThevaluesofrofthisequationarethedistancesRPandRQ,whichareequalinmagnitudebutoppositeinsign.Theconditionforthisisthecoefficientofr=0.
Eliminatingcosθandsinθ,from(4.15)and(4.16),weget
Addinggx1+fy1+ctobothsides,weget
ThisistherequiredequationofthechordPQintermsofitsmiddlepoint(x1,y1).ThisequationcanbeexpressedintheformT=S1whereT=xx1+yy1+g(x
+x1)+f(y+y1)+cand
Note4.12.1:Tistheexpressionwehaveintheequationsofthetangent(x1,y1)tothecircleS:x2+y2+2gx+2fy+c=0andS1istheexpressionwegetbysubstitutingx=x1andy=y1intheleft-handsideofS=0.
4.13COMBINEDEQUATIONOFAPAIROFTANGENTSFROM(x1,y1)TOTHECIRCLEx2+y2+
2gx+2fy+c=0
Lettheequationofachordthrough(x1,y1)be
Anypointonthislineis(x1+rcosθ,y1+rsinθ).Ifthispointliesonthecirclex2+y2+2gx+2fy+c=0,then
Ifchord(4.17)isatangenttocircle(4.18),thenthetwovaluesofrofthisequationareequal.Theconditionforthisis
Butfrom(4.17)
Substitutingthisin(4.19),weget
Thisequationisthecombinedequationofthepairoftangentsfrom(x1,y1).
4.14PARAMETRICFORMOFACIRCLE
x=acosθ,y=asinθsatisfytheequationx2+y2=a2.Thispointisdenotedby‘θ’,whichiscalledaparameterforthecirclex2+y2=a2.
4.14.1EquationoftheChordJoiningthePoints‘θ’and‘ϕ’ontheCircleandtheEquationoftheTangentatθ
Thetwogivenpointsare(acosθ,asinθ)and(acosϕ,asinϕ).Theequationofthechordjoiningthesetwopointsis
Thischordbecomesthetangentat‘θ’ifϕ=0.Therefore,theequationofthetangentat‘θ’isxcosθ+ysinθ=a.
ILLUSTRATIVEEXAMPLES
Example4.1
Findtheequationofthecirclewhosecentreis(3,−2)andradius3units.
Solution
Theequationofthecircleis
Example4.2
Findtheequationofthecirclewhosecentreis(a,−a)andradius‘a’.
Solution
Thecentreofthecircleis(a,−a).Theradiusofthecircleisa.Theequationofthecircleis(x−a)2+(y+a)2=a2(i.e.)x2−2ax+a2+y2+2ay+a2=a2(i.e.)x2+y2−2ax+2ay+a2=0.
Example4.3
Findthecentreandradiusofthefollowingcircles:
i. x2+y2−14x+6y+9=0
ii. 5x2+5y2+4x−8y−16=0
Solution
i.
ii.
Example4.4
Findtheequationofthecirclewhosecentreis(2,−2)andwhichpassesthroughthecentreofthecirclex2+y2−6x−8y−5=0
Solution
Thecentreoftherequiredcircleis(2,−2).Thecentreofthecirclex2+y2−6x−8y−5=0is(3,4).Theradiusoftherequiredcircleisgivenbyr2=(2−3)2+(−2−4)2=1+36=37.
Therefore,theequationoftherequiredcircleis(x−2)2+(y+2)2=37(i.e.)x2+y2−4x+4y−29=0
Example4.5
Showthattheline4x−y=17isadiameterofthecirclex2+y2−8x+2y=0.
Solution
Thecentreofthecirclex2+y2−8x+2y=0is(4,−1).Substitutingx=4andy=−1intheequation4x−y=17,weget16+1=17,whichistrue.Therefore,theline4x−y=17passesthroughthecentreofthegivencircle.Hence,thegivenlineisadiameterofthecircle.
Example4.6
Provethatthecentresofthecirclesx2+y2+4y+3=0,x2+y2+6x+8y−17=0andx2+y2−30x−16y−42=0arecollinear.
Solution
ThecentresofthethreegivencirclesareA(0,−2),B(−3,−4)andC(15,8).
TheslopeofABis
TheslopeofBCis
SincetheslopesABandBCareequalandBisacommonpoint,thepointsA,BandCarecollinear.
Example4.7
Showthatthepoint(8,9)liesonthecirclex2+y2−10x−12y+43=0andfindtheotherendofthediameterthrough(8,9).
Solution
Substitutingx=8andy=9inx2+y2−10x−12y+43=0,weget64+81−80−108+43=0(i.e.)188−188=0,whichistrue.Therefore,thepoint(8,9)liesonthegivencircle.Thecentreofthiscircleis(5,6).Let(x,y)betheotherendofthediameter.
Hence,theotherendofthediameteris(2,3).
Example4.8
Findtheequationofthecirclepassingthroughthepoints(1,1),(2,−1)and(3,2).
2).
Solution
Lettheequationofthecirclebex2+y2+2gx+2fy+c=0.Thecirclepassesthroughthepoints(1,1),(2,−1)and(3,2).
Fromequation(4.20),
Fromequation(4.20),−5−1+c=−2⇒c=4
Therefore,theequationofthecircleisx2+y2−5x−y+4=0.
Example4.9
Showthatthepoints(3,4),(0,5)(−3,−4)and(−5,0)areconcyclicandfindtheradiusofthecircle.
Solution
Lettheequationofthecirclebex2+y2+2gx+2fy+c=0.Thispassesthroughthepoints(3,4),(0,5)and(−3,−4).Therefore,
Fromequation(4.28),f=0Fromequation(4.25),c=25Hence,theequationofthecircleisx2+y2−25=0(4.30)Substitutingx=−5andy=0inequation(4.30),weget0+25−25=0,whichistrue.Therefore,(−5,0)alsoliesonthecircle.Hence,thefourgivenpointsareconcyclic.Thecentreofthecircleis(0,0)andtheradiusis5units.
Example4.10
Findtheequationofthecirclewhosecentreliesonthelinex=2yandwhichpassesthroughthepoints(−1,2)and(3,−2).
Solution
Lettheequationofthecirclebex2+y2+2gx+2fy+c=0.Thispassesthroughthepoints(−1,2)and(3,−2).Therefore,
Subtracting,weget
Substitutingthisinequation(4.33),weget
−2f+f=1⇒f=−1∴g=−2
From(4.31),4−4+c=−5⇒c=⇒−5Hence,theequationofthecircleisx2+y2−4x−2y−2y−5=0.
Example4.11
Findtheequationofthecirclecuttingoffintercepts4and6onthecoordinateaxesandpassingthroughtheorigin.
Solution
Lettheequationofthecirclebex2+y2+2gx+2fy+c=0.Thispassesthroughthepoints(0,0),(4,0)and(0,6).
Thus,theequationofthecircleisx2+y2−4x−6y=0.
Example4.12
Findtheequationofthecircleconcentricwithx2+y2−8x−4y−10=0andpassingthroughthepoint(2,3).
Solution
Twocirclesaresaidtobeconcentriciftheyhavethesamecentre.Therefore,theequationoftheconcentriccircleisx2+y2−8x−4y+k=0.Thiscirclepassesthrough(2,3).
∴4+9−16−12+k=0∴k=15Hence,theequationoftheconcentriccircleisx2+y2−8x−4y+15=0.
Example4.13
Findtheequationofthecircleonthejoiningthepoints(4,7)and(−2,5)astheextremitiesofadiameter.
Solution
Theequationoftherequiredcircleis(x−x1)(x−x2)+(y−y1)(y−y2)=0
Example4.14
Theequationoftwodiametersofacircleare2x+y−3=0andx−3y+2=0.Ifthecirclepassesthroughthepoint(−2,5),finditsequation.
Solution
Thecentreofthecircleisthepointofintersectionofthediameter.
Thecentreofthecircleisthepointofintersectionofthediameter.
Addingthesetwoequations,weget7x=7.∴x=1From(4.38),y=1.Hence,thecentreofthecircleis(1,1)andradiusis
Therefore,theequationofthecircleis
Example4.15
Findthelengthofthetangentfromthepoint(2,3)tothecirclex2+y2+8x+4y+8=0.
Solution
ThelengthofthetangentfromP(x1,y1)tothecirclex2+y2+2gx+2fy+c=0
isgivenby Here,thelengthofthetangentfrom
P(2,3)tothegivencircleis
Example4.16
Determinewhetherthefollowingpointslieoutside,onorinsidethecirclex2+y2
−4x+4y−8=0:A(0,1),B(5,9),C(−2,3).
Solution
Theequationofthecircleisx2+y2−4x+4y−8=0.
Therefore,pointAliesinsidethecircle.PointsBandClieoutsidethecircle.
Example4.17
Findtheequationofthetangentatthepoint(2,−5)onthecirclex2+y2−5x+y−14=0.
Solution
Givenx2+y2−5x+y−14=0
Therefore,theequationofthetangentis
Example4.18
Findthelengthofthechordofthecirclex2+y2−4x−6y−3=0giventhat(1,1)isthemidpointofachordofthecircle.
Solution
Centreofthecircleis(2,3)andradius PointM(1,1)isthe
midpointofthechordAB.
Therefore,thelengthofthechord units.
Example4.19
Showthatthecirclesx2+y2−2x+6y+6=0andx2+y2−5x+6y+15=0toucheachotherinternally.
Solution
Forthecirclex2+y2−2x+6y+6=0,
centreisA(1,−3)andradius units
Forthecirclex2+y2−5x+6y+15=0,
centreis andradius
Distancebetweenthecentresis
Thus,thedistancebetweenthecentresisequaltothedifferenceinradii.Hence,thetwocirclestoucheachotherinternally.
Example4.20
TheabscissaofthetwopointsAandBaretherootsoftheequationx2+2x−a2
=0andtheordinatesaretherootsoftheequationy2+4y−b2=0.FindtheequationofthecirclewithABasitsdiameter.Alsofindthecoordinatesofthecentreandthelengthoftheradiusofthecircle.
Solution
Lettherootsoftheequationx2+2x−a2=0beαandβ.Then
Letγ,δbetherootsoftheequationx2+4y−b2=0.Then
ThecoordinatesofAandBare(α,γ)and(β,δ).TheequationofthecircleonthelinejoiningthepointsAandBastheendsofadiameteris(x−α)(x−β)+(y−γ)(y−δ)=0.
Thecentreofthecirclein(−1,−2)andtheradius
Example4.21
Findtheequationofacirclethatpassesthroughthepoint(2,0)andwhosecentreisthelimitpointoftheintersectionofthelines3x+5y=1and(2+c)x+5c2y=1asc→1.
Solution
Thecentreofthecircleisthepointofintersectionofthelines
Asc→1,thex-coordinateofthecentreis .
From(4.42),
Hence,thecentreofthecircleis
Radiusisthelengthofthelinejoiningthepoints(2,0)and
Therefore,theequationofthecirclesis
Example4.22
Findthelengthinterceptedonthey-axisbythechordofthecirclejoiningthepoints(−4,3)and(12,−1)asdiameter.
Solution
Theequationofthecircleis
Ify1andy2arethey-coordinatesofthepointofintersectionofthecircleandy-axis,then
Example4.23
Therodswhoselengthsareaandbslidealongthecoordinateaxesinsuchawaythattheirextremitiesareconcyclic.Findthelocusofthecentreofthecircle.
Solution
LetABandCBbetheportionofx-axisandy-axis,respectively,interceptedbythecircle.LetP(x1,y1)bethecentreofthecircle.DrawPLandPMperpendiculartox-axisandy-axis,respectively.Then,bysecondproperty
ThelocusofP(x1,y1)is4(x2−y2)=a2−b2.
Example4.24
Showthatthecirclesx2+y2−2x−4y=0andx2+y2−8y−4=0toucheachother.Findthecoordinatesofthepointofcontactandtheequationofthecommontangents.
Solution
ThecentresofthesetwocirclesareC1(1,2)andC2(0,4).Theradiiofthetwocirclesare
Thedistancebetweenthecentresis
∴r1−r2=C1C2.Hence,thecirclestoucheachotherinternally.ThepointofcontactCdividesC1C2internallyintheratio1:1.
IfCisthepoint(x1,y1)then
∴C(2,0)isthepointofcontact.Theslopeof
Hence,theslopeofthecommontangentis1/2.Theequationofthecommon
tangentis
Example4.25
ShowthatthegeneralequationofthecirclethatpassesthroughthepointA(x1,
y1)andB(x2,y2)maybewrittenas
Solution
LetA(x1,y1)andB(x2,y2)bethetwopointsonthecircumferenceofthecircleandA(x1,y1)beanypointonthecircumference.
Let .TheslopeofAPandBPare and
Example4.26
Showthatifthecirclex2+y2=a2cutsoffachordoflength2bontheliney=mx+c,thenc2=(1+m)2(a2−b2).
Solution
Givenx2+y2=a2.Thecentreofthecircleis(0,0).Radius=r=a.DrawOLperpendiculartoAB.Then,ListhemidpointofAB.
Example4.27
Apointmovessuchthatthesumofthesquaresofthedistancesfromthesidesofasquareofsideunityisequaltoa.Showthatthelocusisacirclewhosecentrecoincideswiththecentreofthesquare.
Solution
Letthecentreofthesquarebetheorigin.LetP(x,y)beanypoint.Then,the
equationofthesidesare
SumoftheperpendiculardistancesfromPonthesidesisequaltoa
Hence,thelocusofPisthecirclex2+y2−1=0.Thecentreofthecircleis(0,0),whichisthecentreofthesquare.
Example4.28
Ifthelinesl1x+m1y+n1=0andl2x+m2y+n2=0cutthecoordinateaxesatconcyclicpoints,provethatl1l2=m1m2.
Solution
Givenl1x+m1y+n1=0.Theinterceptsofthelineontheaxisare
IfthelinemeetstheaxesatL1andM1,then Ifthesecond
linemeetstheaxesatL2andM2,then
Example4.29
Showthatthelocusofapointwhoseratioofdistancesfromtwogivenpointsisconstantisacircle.Hence,showthatthecirclecannotpassthroughthegivenpoints.
Solution
LetthetwopointsAandBbechoseninthex-axisandthemidpointofABbe(0,0).ThenletA(a,0)andB(−a,0).GiventhatPA=K·PB⇒PA2=K2PB2wherekisaconstant.
(x−a)2+(y−0)2=K2[(x+a)2+y2]
Inthisequation,thecoefficientsofx2andy2arethesameandthereisnoxyterm.Therefore,thelocusofPisacircle.IfA(a,0)liesonthiscircle,thenO=K2[4a2]⇒a=0ork=0,whicharenotpossible.Therefore,thepointAdoesnotlieonthecircle.Similarly,thepointB(−a,0)alsodoesnotlieonthecircle.
Example4.30
Findtheequationofthecirclewhoseradiusis5andwhichtouchesthecirclex2
+y2−2x−4y−20=0atthepoint(5,5).
Solution
Givenx2+y2−2x−4y−20=0.
Centreis(1,2)andradius=
Letthecentreoftherequiredcirclebe(x1,y1).ThepointofcontactisthemidpointofAB.
∴x=9andy=8Thus,Bis(9,8).Hencetheequationoftherequiredcircleis
Example4.31
OneofthediametersofthecirclecircumscribingtherectangleABCDis4y=x+7.IfAandBarethepoints(−3,4)and(5,4),respectively,findtheareaoftherectangleABCD.
Solution
LetP(x1,y1)bethecentreofthecircleand4y=x+7betheequationofthediameterofBD.
ThemidpointofACis(1,1).TheslopeofABis0.Therefore,theslopeofPLis∞.
Hence,theareaoftherectangleABCD=8×4=32sq.cm.
Example4.32
Findtheequationofthecircletouchingthey-axisat(0,3)andmakinganinterceptof8cmonthex-axis.
Solution
Lettheequationofthecirclebex2+y2+2gx+2fy+c=0.Centreis(−g,−f),Thus,−f=3orf=−3.Whenthecirclemeetsthex-axis,y=0.
Hence,theequationofthecircleisx2+y2±10x−6y+9=0.
Example4.33
Findtheequationofthecirclepassingthroughthepoint(−4,3)andtouchingthelinesx+y=2andx−y=2.
Solution
x+y=2andx−y=2intersectatthepoint(2,0).
Moreover,theselinesareperpendicularandtheirslopesare1and−1.So,theymake45°and135°withthex-axis.Henceoneofthebisectorsisthex-axisandcentreliesononeofthebisectors.Ifx2+y2+2gx+2fy+c=0istheequationofthecircle,thenf=0.Alsotheperpendiculardistancefrom(−g,0)tothetangentsisequaltotheradius.
Since(−4,3)liesonthecircle16+9−8g+c=0
Hence,equation(4.46)becomesg2−4g−4−16g+50=0org2−20g+46=0.
Thus,therearetwocircleswhoseequationsaregivenby
Example4.34
Aisthecentreofthecirclex2+y2−2x−4y−20=0.IfthetangentsdrawnatthepointsB(1,7)andD(4,−2)onthecirclemeetatthepointC,thenfindtheareaofthequadrilateralABCD.
Solution
x2+y2−2x−4y−20=0
Centreofthiscircleis(1,2)
Theequationsoftangentsat(1,7)and(4,−2)tothecirclearex+7y−(x+1)−2(y+7)−20=0(i.e.)5y−35=0⇒y=7and
4x−2y−(x+4)−2(y−2)−20=0(i.e.)3x−4y−20=0
Sincey=7,x=16.Hence,thepointCis(16,7).AreaofthequadrilateralABCD=2×areaofΔABC
Example4.35
FromthepointA(0,3)onthecirclex2+4x+(y−3)2=0,achordABisdrawnandextendedtoapointMsuchthatAM=AB.FindtheequationofthelocusofM.
Solution
AM=2.AB
Hence,BisthemidpointofAM.ThenthecoordinatesofBare
ThispointBliesonthecirclex2+4x+(y−3)2=0.
Therefore,thelocusof(x1,y1)isx2+y2+8x−6y+9=0.
Example4.36
ABisadiameterofacircle,CDisachordparalleltoABand2CD=AB.ThetangentatBmeetsthelineACproducedatE.ProvethatAE=2.AB.
Solution
Lettheequationofthecirclebex2+y2=a2andPQbethediameteralongthex-axis.CDisparalleltoAB.LetAB=2aandpointsAandBbe(a,0)and(−a,0),respectively.Also
Example4.37
Findtheareaofthetriangleformedbythetangentsfromthepoint(h,k)tothecirclex2+y2=a2andtheirchordofcontact.
Solution
Theequationofthecircleisx2+y2=a2.LetABbethechordofcontactoftangentsfromC(x1,y1).ThentheequationofABisxx1+yy1=a2.
WeknowthatOCisperpendiculartoAB.LetABandOCmeetatL.
TheperpendiculardistancefromConAB
Example4.38
Letacirclebegivenby2x(x−a)+y(2y−b)=0,(a,b≠0).Findtheconditiononaandbiftwochordseachintersectedbythex-axiscanbedrawntothecircle
from
Solution
Thechordisbisectedbythex-axis.Letthemidpointofthechordbe(h,0).Theequationofthechordis
Thischordpassesthrough .
Sincethechordmeetsthex-axisattworeals,Discriminant>0
Example4.39
Findtheconditionthatthechordofcontactfromapointtothecirclex2+y2=a2
subtendsarightangleatthecentreofthecircle.
Solution
Theequationtothechordofcontactfrom(x1,y1)tothecircle
ThenthecombinedequationtoOAandOBisgotbyhomogenizingequation(4.48)withthehelpofequation(4.49).
ThecombinedequationofOAandOBis
Since ,coefficientofx2+coefficientofy2=0.
Example4.40
Ify=mxbetheequationofachordofthecirclewhoseradiusisa,theoriginbeingoneoftheextremitiesofthechordandtheaxisbeingadiameterofthecircle,provethattheequationofacircleofwhichthischordisadiameteris(1+m2)(x2+y2)−2a(x+my)=0.
Solution
Letabetheradiusofthecircle.Thus(a,0)isthecentreofthecircle.Theequationofthecircleis
(x−a)2+y2=a2⇒x2+y2−2ax=0
Wheny=mxmeetsthecirclex2+m2x2−2ax=0.
Therefore,theextremitiesofthischordare(0,0)and Then,the
equationofthecirclewiththechordasadiameteris
Example4.41
Findtheequationtothecirclethatpassesthroughtheoriginandcutsoffequalchordsoflengthafromthestraightlinesy=xandy=−x.
Solution
Letthelinesy=xandy=−xmeetthecircleatP,P′andQ,Q′,respectively.
ThenOP=OQ=a=OP′=OQ′.ThecoordinatesofPandP′are
SimilarlythecoordinatesofQandQ′are
Therearefourcirclespossiblehavingcentresat
Hence,theequationsofthefourcirclesaregivenby
Example4.42
Findthelocusofthemidpointofchordsofthecirclex2+y2=a2,whichsubtendsarightangleatthepoint(c,0).
Solution
SinceABsubtends90°atC(c,0),PA=PB=PC.LetPbethepoint(x1,y1).
SincePisthemidpointofthechordAB,CP⊥AP
Since ,PC=AP.
Thelocusof(x1,y1)is2(x2+y2)−2cx1+(c2−a2)=0.
Example4.43
Findtheequationsofthecirclesthattouchthecoordinateaxesandthelinex=a.
Solution
y=0,x=0andx=aarethetangentstothecircle.Therearetwocirclesasshowninthefigure.
Thecentresare andradius .Theequationsofthecirclesare
Example4.44
Findtheshortestdistancefromthepoint(2,−7)tothecirclex2+y2−14x−10y−151=0.
Solution
x2+y2−14x−10y−151=0
Centeris(7,5)
Centeris(7,5)
Radius=
TheshortestdistanceofthepointPfromthecircle=∣CP−r∣
Example4.45
Letα,βandγbetheparametricanglesofthreepointsP,QandR,respectively,onthecirclex2+y2=a2andAbethepoint(−a,0).Ifthelengthofthechords
AP,AQandARareinAPthenshowthat arealsoinAP.
Solution
LetP(acosα,asinα),Q(acosβ,asinβ),R(acosr,asinr)Ais(a,0)
ThelengthsofchordsAP,AQ,ARareinAP.
Example4.46
LetS=x2+y2+2gx+2fy+c=0.Findthelocusofthefootoftheperpendicularfromtheoriginonanychordofthecirclethatsubtendsarightangleattheorigin.
Solution
LettheequationofthelineABbe
lx+my=1(4.50)
Let(x1,y1)bethemidpointofAB.
LetP(x1,y1)bethefootoftheperpendicularfromtheoriginonAB.Then,sinceOPisperpendiculartoAP.
Since(x1,y1)liesonthelinelx+my=1wehave
lx1+my1=1(4.52)
ThecombinedequationoflinesOAandOBisgotbyhomogenizingtheequationofthecirclex2+y2+2gx+2fy+c=0withthelinelx+my=1.
Since ,theconditioniscoefficientofx2+coefficientofy2=0.Hence,
Thelocusof(x1,y1)is2(x2+y2)+2gx+2fy+c=0.
Example4.47
Pisthepoint(a,b)andQisthepoint(b,a).FindtheequationofthecircletouchingOPandOQatPandQwhereOistheorigin.
Solution
Lettheequationofthecirclebe
LetC(−g,−f)bethecentreofthecircle.
WeknowthatPQisthechordofthecontactfromOandOCisperpendiculartoPQ.∴SlopeofPQ×slopeofOC=−1
TheequationofOPis
SinceCPisperpendiculartoOP,ristheperpendiculardistancefromConOP.
Thepoint(a,b)liesonthecircle(4.53).
Hencetheequationofthecircleis
Example4.48
Acircleofcircumradius3kpassesthroughtheoriginandmeetstheaxesatAandB.ShowthatthelocusofthecentroidofΔOABisthecirclex2+y2=4K2.
Solution
LetAandBbethepoints(a,0)and(0,b),respectively.Let(x1,y1)bethe
centroidofΔOAB.Thensince ,ABisadiameterofthecircle.
LetthecentroidofΔOABbe(x1,y1).Then ora=3x1andb=
3y1.Substitutingthisin(4.55),weget .Thelocusof(x1,
y1)isx2+y2=4k2.
Example4.49
Avariablelinepassesthroughafixedpoint(a,b)andcutsthecoordinateaxesat
thepointsAandB.ShowthatthelocusofthecentreofthecircleABis
Solution
LetABbeavariablelinewhoseequationbe
Thispassesthroughthepoint(a,b).
Since ABisadiameterofthecircumcircleofΔOAB.Itscentreis
If(x1,y1)bethecircumcentre,then
∴α=2x1andβ=2y1
Hence,from(4.55),weget
Thelocusof(x1,y1)is
Example4.50
If4l2−5m2+6l+1=0thenshowthatthelinelx+my+1=0touchesafixedcircle.Findthecentreandradiusofthecircle.
Solution
Lettheline
touchthecircle
Thentheperpendiculardistancefrom(h,k)toline(4.56)isequaltotheradius.
Buttheconditionisgivenby
Identifying(4.58)and(4.59),weget
Hence,thelinetouchesthefixedcircle(x−3)2+y2=5orx2+y2−6x+4=0
whosecentreis(3,0)andradiusis
Exercises
1. Findtheequationofthefollowingcircles:
i. centre(2,−5)andradius5unitsii. centre(−2,−4)andradius10unitsiii. centre(a,b)andradius(a+b)
Ans.:(i)x2+y2−4x+10y+4=0
Ans.:(ii)x2+y2+4x+8y−80=0
Ans.:(iii)x2+y2−2ax−2by=0
2. Findthecentreandradiusofthefollowingcircles:
i. x2+y2−22x−4y+25=0
ii. 4(x2+y2)−8(x−2y)+19=0
iii. 2x2+2y2+3x+y+1=0
Ans.:(i)(11,2),10
Ans.:(ii)(1,−2),
Ans.:
3. Findtheequationofthecirclepassingthroughthepoint(2,4)andhavingitscentreonthelinesx−y=4and2x+3y=8.
Ans.:x2+y2−8x−4=0
4. Findtheequationofthecirclewhosecentreis(−2,3)andwhichpassesthroughthepoint(2,−2).
Ans.:x2+y2+4x−6y−28=0
5. Showthattheline4x−y=17isadiameterofthecirclex2+y2−8x+2y=0.
6. Theequationofthecircleisx2+y2−8x+6y−3=0.Findtheequationofitsdiameterparallelto2x−7y=0.Alsofindtheequationofthediameterperpendicularto3x−4y+1=0.
Ans.:2x−7y−29=04x+3y−7=0
7. Findtheequationofthecirclepassingthroughthefollowingpoints:i. (2,1),(1,2),(8,9)ii. (0,1),(2,3),(−2,5)iii. (5,2),(2,1),(1,4)
Ans.:x2+y2−10x−10y−25=0
Ans.:3x2+3y2+2x−20y+17=0
Ans.:x2+y2−6x−6y+13=0
8. Findtheequationofthecirclethroughthepoints(1,0)and(0,1)andhavingitscentreonthelinex+y=1.
Ans.:x2+y2−x−y=0
9. Findtheequationofthecirclepassingthroughthepoints(0,1)and(4,3)andhavingitscentreontheline4x−5y−5=0.
Ans.:x2+y2−5x−2y+1=0
10. Twodiametersofacircleare5x−y=3and2x+3y=8.Thecirclepassesthroughthepoint(−1,7).Finditsequation.
Ans.:x2+y2−2x−4y=164
11. Findtheequationofthecirclecircumscribingthetriangleformedbytheaxesandthestraightline3x+4y+12=0.
Ans.:x2+y2+4x+3y=0
12. Showthatthepoints(−1,2),(−2,4),(−1,3)and(2,0)areonacircleandfinditsequation.13. Ifthecoordinatesoftheextremitiesofthediameterofacircleare(3,5)and(−7,−5),findthe
equationofthecircle.
Ans.:x2+y2+4x−3y=0
14. Findtheequationofthecirclewhenthecoordinatesoftheextremitiesofoneofitsdiametersare(4,1)and(−2,–7).
Ans.:x2+y2−2x+6y−15=0
15. Ifoneendofthediameterofthecirclex2+y2−2x+6y−15=0is(4,1),findthecoordinatesoftheotherend.
Ans.:(−2,−7)
16. Provethatthetangentsfrom(0,5)tothecirclesx2+y2+2x−4=0andx2+y2−y+1=0areequal.
17. Findtheequationofthecirclepassingthroughtheoriginandhavingitscentreat(3,4).Alsofindtheequationofthetangenttothecircleattheorigin.
Ans.:x2+y2−6x−8y=0,3x+4y=0
18. Findtheslopeoftheradiusofthecirclex2+y2=25throughthepoint(3,−4)andhencewritedowntheequationofthetangenttothecircleatthepoint.Whataretheinterceptsmadebythistangentonthex-axisandy-axis?
Ans.:
19. Onevertexofasquareistheoriginandtwoothersare(4,0)and(0,4).Findtheequationofthecirclecircumscribingthesquare.Alsofindtheequationofthetangenttothiscircleattheorigin.
Ans.:x2+y2−4x−4=0,x+y=0
20. Acirclepassesthroughtheoriginandthepoints(6,0)and(0,8).Finditsequationandalsotheequationofthetangenttothecircleattheorigin.
Ans.:x2+y2−6x−8y=0,3x+4y=0
21. AandBaretwofixedpointsonaplaneandthepointPmovesontheplaneinsuchawaythatPA=2PBalways.ProveanalyticallythatthelocusofPisacircle.
22. Doesthepoint(2,1)lie(i)on,(ii)insideor(iii)outsidethecirclex2+y2−4x−6y+9=0?
23. Showthatthecirclesx2+y2−2x+2y+1=0andx2+y2+6x−4y−3=0toucheachotherexternally.
24. Provethatthecentresofthethreecirclesx2+y2−2x+6y+1=0,x2+y2+4x−12y−9=0and
x2+y2=25lieonthesamestraightline.Whatistheequationofthisline?Ans.:3x+4y=0
25. Provethatthetwocirclesx2+y2+2ax+c2=0andx2+y2+2by+c2=0toucheachotherif
26. Showthatthecirclesx2+y2−4x+2y+1=0andx2+y2−12x+8y+43=0toucheachotherexternally.
27. Showthatthecirclesx2+y2=400andx2+y2−10x−24y+120=0touchoneanother.Findthecoordinatesofthepointofcontact.
Ans.:
28. Findthelengthofthetangentfromtheorigintothecircle4x2+4y2+6x+7y+1=0.
29. Showthatthecirclesx2+y2−26x−19=0andx2+y2+3x−8y−43=0touchexternally.Findthepointofcontactandthecommontangent.
30. Apointmovessothatthesquareofitsdistancefromthebaseofanisoscelestriangleisequaltotherectanglecontainedbyitsdistancesfromtheequalsides.Provethatthelocusisacircle.
31. Provethatthecentresofthecirclesx2+y2=1,x2+y2+4x+8y−1=0andx2+y2−6x−12y+1=0arecollinear.
32. Provethattheconstantintheequationofthecirclex2+y2+2gx+2fy+c=0isequaltotherectangleunderthesegmentsofthechordsthroughtheorigin.
33. Findtheequationofthelocusofapointthatmovesinaplanesothatthesumofthesquaresfromtheline7x−4y−10=0and4x+7y+5=0isalwaysequalto3.
Ans.:13x2+13y2−20x+30y−14=0
34. Showthatthecirclesx2+y2−10x+4y−20=0andx2+y2+14x−6y+22=0toucheachother.Findtheequationoftheircommontangentatthepointofcontactandalsothepointofcontact.
Ans.:
35. LandMarethefeetoftheperpendicularfrom(c,0)onthelinesax2+2hxy+by2=0.ShowthattheequationofLMis(a+b)x+2hy+bc=0.
36. Acirclehasradius3unitsanditscentreliesontheliney=x−1.Findtheequationofthecircleifitpassesthrough(−1,3).
37. Findtheequationofthecircleonthelinejoiningthepoints(−4,3)and(12,−1).Findalsotheinterceptsmadebyitonthey-axis.
Ans.:
38. Showthatthepoints liesoutsidethecircle3x2+3y2−5x−6y+4=0.
39. Findtheconditionthatthelinelx+my+n=0touchesthecirclex2+y2=a2.Findalsothepointofcontact.
Ans.:
40. Findtheequationofthecirclepassingthroughthepoint(3,5)and(5,3)andhavingitscentreontheline2x+3y−1=0.
Ans.:5x2+3y2−14x−14y−50=0
41. ABCDisasquarewhosesideisa.TakinglineAOastheaxisofcoordinates,provethatthe
equationofthecircumcircleofthesquareisx2+y2−ax−ay=0.42. Findtheequationofthecirclewithitscentreontheline2x+y=0andtouchingthelines4x−3y
+10=0and4x−3y−3=0.
Ans.:x2+y2−2x+4y−11=0
43. Findtheequationofthecirclethatpassesthroughthepoint(1,1)andtouchesthecirclex2+y2+4x−6y−3=0atthepoint(2,3)onit.
Ans.:x2+y2+x−6y+3=0
44. Provethatthetangenttothecirclex2+y2=5atthepoint(1,−2)alsotouchesthecirclex2+y2−8x+6y+20=0andfinditspointofcontact.
Ans.:(3,−1)
45. AvariablecirclepassesthroughthepointA(a,b)andtouchesthex-axis.Showthatthelocusofthe
otherendofthediameterthroughAis(x−c)2=4by.46. FindtheequationofthecirclepassingthroughthepointsA(−5,0),B(1,0),andC(2,1)andshow
thattheline4x−3y−5=0isatangenttotheline.47. Findtheequationofthecirclethroughtheoriginandthroughthepointofcontactofthetangents
fromtheorigintothecircle.
Ans.:2x2+2y2−11x−13y=0
48. Thecirclex2+y2−4x−4y+4=0isinscribedinatrianglethathastwoofitssidesalongthe
coordinateaxes.Thelocusofthecircumferenceofthetriangleis Findk.
Ans.:k=1
49. Acircleofdiameter13mwithcentreOcoincidingwiththeoriginofcoordinateaxeshasdiameterABonthex-axis.IfthelengthofthechordACbe5m,findtheareaofthesmallerportionboundedbetweenthecirclesandthechordAC.
Ans.:1.9m2.
50. Findtheradiusofthesmallestcirclethattouchesthestraightline3x−y=6at(1,−3)andalsotouchestheliney=x.
Ans.:
51. If formdistinctpointsonacircleshowthatm1,m2,m3,m4=1.
52. Ifthelinexcosα+ysinα=ρcutsthecirclex2+y2=a2inMandN,thenshowthatthecircle
whosediameterisMNisx2+y2−a2−2ρ(xcosα+ysinα−ρ)=0.
53. Showthatthetangentsdrawnfromthepoint(8,1)tothecirclex2+y2−2x−4y−20=0areperpendiculartoeachother.
54. Howmanycirclescanbedrawneachtouchingallthethreelinesx+y=1,y=x+1and7x−y=6?Findthecentreandradiusofallthecircles.
Ans.:
55. Findthepointsonthelinex−y+1=0,thetangentsfromwhichtothecirclex2+y2−3x=0areoflength2units.
Ans.:
56. Onthecircle16x2+16y2+48x−3y−43=0,findthepointnearesttotheline8x−4y+73=0andcalculatethedistancebetweenthispointandtheline.
Ans.:
57. Findtheequationsofthelinestouchingthecirclex2+y2+10x−2y+6=0andparalleltotheline2x+y−7=0.
Ans.:2x+y−1=0,2x+y+19=0
58. Findtheequationofthecirclewhosediameteristhechordofintersectionofthelinex+3y=6
andthecurve4x2+9y2=36.
Ans.:5(x2+y2)−12x−16y+12=0
59. Findtheequationforthecircleconcentricwiththecirclex2+y2−8x+6y−5=0andpassesthroughthepoint(−2,7).
Ans.:x2+y2−8x+6y−27=0
60. Findtheequationofthecirclethatcutsoffintercepts−1and−3onthex-axisandtouchesthey-
axisatthepoint
Ans.:
61. Findthecoordinatesofthepointofintersectionoftheline5x−y+7=0andthecirclex2+y2+3x−4y−9=0.Alsofindthelengthofthecommonsegment.
Ans.:
62. Theline4x+3y+k=0isatangenttothecirclex2+y2=4.Findthevalueofk.Ans.:k=±10
63. Findtheequationsoftangentstothecirclex2+y2−6x+4y−17=0thatareperpendicularto3x−4y+5=0.
Ans.:4x+3y+19=0,4x+3y−31=0
64. Findtheequationoftangentstothecirclex2+y2−14x+y−5=0atthepointswhoseabscissais10.
Ans.:3x+7y−93=0,3x−7y−64=0
65. Showthatthecirclesx2+y2−4x+6y+8=0andx2+y2−10x−6y+14=0toucheachother.Findthepointofcontact.
Ans.:(3,−1)
66. Showthatthetangenttothecentrex2+y2=0atthepoint(1,−2)alsotouchesthecirclex2+y2−8x+6y+20=0.Findthepointofcontact.
Ans.:(3,−1)
67. AstraightlineABisdividedatCsothatAB=3CB.CirclesaredescribedonACandCBasdiametersandacommontangentmeetsABproducedatD.ShowthatBDisequaltotheradiusofthesmallercircle.
68. Thelines3x−4y+4=0and6x−8y−7=0aretangentstothesamecircle.Findtheradiusofthiscircle.
Ans.:
69. Fromtheorigin,chordsaredrawntothecircle(x−1)2+y2=1.Findtheequationofthelocusofthemidpointofthesechords.
Ans.:x2+y2−x=0
70. Findtheequationsofthepairoftangentstothecirclex2+y2−2x+4y=0from(0,1).
Ans.:2x2−2y2+3xy−3x+4y−2=0
71. Ifthepolarofpointsonthecirclex2+y2=a2withrespecttothecirclex2+y2=b2touchthe
circlex2+y2=c2,showthata,bandcareinGP.
72. Ifthedistancesoforigintothecentresofthreecirclesx2+y2−2λx=c2whereλisavariableandcisaconstantareinG.P,provethatthelengthofthetangentdrawntothemfromanypointonthe
circlex2+y2=c2areinG.P.
73. Atangentisdrawntoeachofthecirclesx2+y2=a2andx2+y2=b2.Ifthetwotangentsaremutuallyperpendicular,showthatthelocusoftheirpointofintersectionisacircleconcentricwiththegivencircles.
74. Ifthepoleofanylinewithrespecttothecirclex2+y2=a2liesonthecirclex2+y2=9a2,then
showthatthelinewillbeatangenttothecircle .
75. Atrianglehastwoofitssidesalongthey-axis,anditsthirdsidetouchesthecirclex2+y2−2ax−
2ay+a2=0.Provethatthelocusofthecircumcentreofthetriangleis2xy−2a(x+y)+a2=0.76. Lines5x+12y−10=0and6x−11y−40=0touchacircleC,ofdiameter6.IfthecentreofC1
liesinthefirstquadrant,findtheequationofcircleC2.whichisconcentricwithC1andcutsinterceptsoflength8ontheselines.
Ans.:
77. Findtheequationofthecirclethattouchesthey-axisatadistanceof4unitsfromtheoriginandcutsoffaninterceptof6unitsfromthex-axis.
Ans.:x2+y2+10x−8y+16=0
78. Findtheequationofthecircleinwhichthelinejoiningthepoints(0,b)and(b,−a)isachordsubtendinganangle45°atanypointonitscircumference
Ans.:x2+y2−2(a+b)x+2(a−b)y+(a2+b2)
79. Fromanypointonagivencircle,tangentsaredrawntoanothercircle.Provethatthelocusofthemiddlepointofthechordofcontactisathirdcircle;thedistancebetweenthecentresofthegivencircleisgreaterthanthesumoftheirradii.
80. Apointmovessothatthesumofthesquaresoftheperpendicularsthatfallfromitonthesidesofanequilateraltriangleisconstant.Provethatthelocusisacircle.
81. AcircleofconstantradiuspassesthroughtheoriginOandcutstheaxesinAandB.Showthatthe
locusofthefootoftheperpendicularfromABis(x2+y2)2(x2+y2)=4r2.
82. Findtheequationoftheimageofthecircle(x−3)2+(y−2)2=1bythemirrorx+y=19.
Ans.:(x−17)2+(y−16)2=1
83. Findthevalueofλforwhichthecirclex2+y2+6x+5+λ(x2+y2−8x+7)=0dwindlesintoapoint.
Ans.:
84. Avariablecirclealwaystouchestheliney=xandpassesthroughthepoint(0,0).Showthatthe
commonchordsofthiscircleandx2+y2+6x+8y−7=0willpassthroughafixedpoint
85. Theequationofthecirclethattouchestheaxesofthecoordinatesandtheline and
whosecentreliesinthefirstquadrantisx2+y2−2cx−2cy+c2=0.Findthevaluesofc.Ans.:(1,6)
86. Aregioninxy-planeisboundedbythecurve andtheliney=0.Ifthepoint(a,a+1)
liesintheinterioroftheregion,findtherangeofa.Ans.:a∈(−1,3)
87. Thepoints(4,−2)and(3,6)areconjugatewithrespecttothecirclex2+y2=24.Findthevalueofb.
Ans.:b=−6
88. Ifthetwocirclesx2+y2+2gx+2fy=0andx2+y2+2g1x+2ƒ1y=0toucheachother,show
thatƒ1g=gƒ1.89. Showthatthelocusofthepointsofchordsofcontactoftangentssubtendingarightangleatthe
centreisaconcentriccirclewhoseradiusis timestheradiusofthegivencircle.Alsoshowthat
thisisalsothelocusofthepointofintersectionofperpendiculartangents.90. Showthatthepoints(xi,yi),i=1,2,3arecollinearifandonlyiftheirpoleswithrespecttothe
circlesx2+y2=a2areconcurrent.91. ThelengthofthetangentsfromtwogivenpointsAandBtoacirclearet1andt2,respectively.If
thepointsareconjugatepoints,showthat
92. Showthattheequationtothepairoftangentsdrawnfromtheorigintothecirclex2+y2+2gx+
2ƒy+c=0is(gx+ƒy)2=(ƒ2+g2).Hencefindthelocusofthecentreofthecircleifthesetangentsareperpendicular.
Ans.:x2+y2=2c
93. ThreesidesofatrianglehavetheequationsLi=y−mrx−cr=0,r=1,2,3.ThenshowthatλL2L3+μL3L1+vL1L2=0whereλ,μ,v≠0istheequationofthecircumcircleofthetriangleifΣλ(m2+m3)=0andΣλ(m2m3−m1)=0.
94. Atriangleisformedbythelineswhosecombinedequationisc(x+y−4)(xy−2x−y+2)=0.
Showthattheequationofitscircumferenceisx2+y2−3x−5y+8=0.
95. Twodistinctchordsdrawnfromthepoint(p,q)onthecirclex2+y2=px+qy,wherepq≠0,are
bisectedbythex-axis.Showthatp2>8q2.
96. Showthatthenumberofpointswithintegralcoordinatesthatareinteriortothecirclex2+y2=16is45.
97. Findthenumberofcommontangentstothecirclesx2+y2−6x−14y+48=0andx2+y2−6x=0.
Ans.:4
98. Thetangentstothecirclex2+y2=4atthepointsAandBmeetatP(−4,0).FindtheareaofthequadrilateralPAOB.
Ans.:
99. Theequationsoffourcirclesare(x±a)2+(y±a)2=a2.Findtheradiusofacircletouchingallthefourcircles.
Ans.:
100. Acircleofradius2touchesthecoordinateaxesinthefirstquadrant.Ifthecirclemakesacompleterotationonthex-axisalongthepositivedirectionofthex-axis,thenshowthattheequationofthe
circleinthenewpositionisx2+y2−4(x+y)−8λx+(2+4π)2=0.101. Twotangentsaredrawnfromtheorigintoacirclewithcentreat(2,−1).Iftheequationofoneof
thetangentsis3x+y=0,findtheequationoftheothertangent.Ans.:x−3y=0
102. Findtheequationofthechordofthecirclex2+y2=a2passingthroughthepoint(2,3)fartherfromthecentre.
Ans.:2x+3y=17
103. Anequilateraltriangleisinscribedinthecirclex2+y2=a2,withthevertexat(a,0).Findtheequationofthesideoppositetothisvertex.
Ans.:2x+a=0
104. AlineisdrawnthroughthepointP(3,1)tocutthecirclex2+y2=9atAandB.FindthevalueofPA·PB.
Ans.:121
105. C1andC2arecirclesofunitradiuswiththeircentresat(0,0)and(1,0),respectively.C3isacircleofunitradius,passingthroughthecentresofthecirclesC1andC2andhavingitscentreabovethex-axis.FindtheequationofthecommontangenttoC1andC3thatpassesthroughC2.
Ans.:
106. Achordofthecirclex2+y2−4x−6y=0passingthroughtheoriginsubtendsanangle
atthepointwherethecirclemeetsthepositivey-axis.Findtheequationofthechord.Ans.:x−2y=0
107. AcirclewithitscentreattheoriginandradiusequaltoameetstheaxisofxatAandB.PandQarerespectivelythepoints(acosα,atanα)and(acosβ,atanβ)suchthatα−β=2γ.Showthatthe
locusofthepointofintersectionofAPandBQisx2+y2−2aytanγ=−2.
108. AcircleC1ofradiustouchesthecirclex2+y2=a2externallyandhasitscentreonthepositivex-
axis.AnothercircleC2ofradiusctouchescircleC1externallyandhasitscentreonthepositivex-axis.Ifa<b<c,showthatthethreecircleshaveacommontangentifa,b,careinGP.
109. Findtheequationsofcommontangentstothecirclesx2+y2+14x−14y+28=0andx2+y2−14x+4y−28=0
Ans.:28y+45y+371=0andy−7=0.
110. Ifacirclepassesthroughthepointsofintersectionofthecoordinateaxeswiththelinex−λy+1=
0(λ≠0)andx−2y+3=0thenλsatisfiestheequation6λ2−7λ+2=0.
111. OAandOBareequalchordsofthecirclex2+y2−2x+4y=0perpendiculartoeachotherand
passingthroughtheorigin.ShowthattheslopesofOAandOBsatisfytheequation3m2−8m−3=0.
112. Findtheequationofthecirclepassingthroughthepoints(1,0)and(0,1)andhavingthesmallestpossibleradius.
Ans.:x2+y2−x−y=0
113. Findtheequationofthecirclesituatedsystematicallyoppositetothecirclex2+y2−2x=0withrespecttothelinex+y=2.
Ans.:x2+y2−4x−2y+4=0
114. OisafixedpointandRmovesalongafixedlineLnotpassingthroughO.IfSistakenonORsuch
thatOR·OS=K2,thenshowthatthelocusofSisacircle.115. Showthatthecircumferenceofthetriangleformedbythelinesax+by+c=0,bx+cy+a=0
andcx+ay+b=0passesthroughtheoriginif(b2+c2)(c2+a2)(a2+b2)=abc(b+c)(c+a)(a+b).
116. Twocirclesaredrawnthroughthepoints(a,5a)and(4a,a)totouchthey-axis.Provethatthey
intersectatanangle
117. ShowthatthelocusofapointPthatmovessothatitsdistancefromthegivenpointOisalwaysinagivenration:1·(n≠−1)toitsdistanceonthelinejoiningthepointsthatdividesthelineOAinthegivenratioasdiameter.
118. Thelines3x−4y+4=0and6x−3y−7=0aretangentstothesamecircle.Findtheradiusofthecircle.
Ans.:
119. Theliney=xtouchesacircleatPsothat whereOistheorigin.Thepoint(−10,2)is
insidethecircleandlengthofthechordontheline Findtheequationofthe
line.
Ans.:x2+y2+18x−2y+32=0
120. Findtheintervalsofvaluesofaforwhichtheliney+x=0bisectstwochordsdrawnfromapoint
tothecircle
121. Showthatallchordsofthecircle3x2−y2−2x+4y=0thatsubtendarightangleattheoriginare
concurrent.Doestheresultholdforthecurve3x2+3y2−2x+4y=0?Ifyes,whatisthepointofconcurrency,andifnot,givethereason.
122. Findtheequationsofthecommontangentstothecirclesx2+y2−14x+6y+33=0andx2+y2
+30x−20y+1=0.Ans.:4x−3y−12=0,24x+7y−22=0
123. Provethattheorthocentreofthetrianglewhoseangularpointsare(acosα,asinα),(acosβ,asinβ)and(acosγ,asinγ)isthepoint[a(cosα+cosβ+cosγ),a(sinα+sinβ+sinγ)].
Chapter5
SystemofCircles
5.1RADICALAXISOFTWOCIRCLES
Definition5.1.1:Theradicalaxisoftwocirclesisdefinedasthelocusofapointsuchthatthelengthsoftangentsfromittothetwocirclesareequal.
ObtaintheequationoftheradicalaxisofthetwocirclesS≡x2+y2+2gx+2fy+c=0andS1≡x2+y2+2g1x+2fy+c1=0.
LetP(x1,y1)beapointsuchthatthelengthsoftangentstothetwocirclesareequal.
Thelocusof(x1,y1)is2(g−g1)x+2(f−f1)y+(c−c1)=0whichisastraightline.Therefore,theradicalaxisoftwogivencircleisastraightline.
Note5.1.1:IfS=0andS1=0aretheequationsoftwocircleswithunitcoefficientsforx2andy2termsthentheequationoftheradicalaxisisS−S1=0.
Note5.1.2:Radicalaxisoftwocirclesisastraightlineperpendiculartothelineofcentres.ThecentresofthetwocirclesareA(−g,−f)andB(−g1,−f1).
Theslopeofthelineofcentresis
Theslopeoftheradicalaxisis
∴m1m2=–1
Therefore,theradicalaxisisperpendiculartothelineofcentres.
Note5.1.3:IfthetwocirclesS=0andS1=0intersectthentheradicalaxisisthecommonchordofthetwocircles.
Note5.1.4:Ifthetwocirclestoucheachother,thentheradicalaxisisthecommontangenttothecircles.
Note5.1.5:Ifacirclebisectsthecircumferenceofanothercirclethentheradicalaxispassesthroughthecentreofthesecondcircle.
Showthattheradicalaxesofthreecirclestakentwobytwoareconcurrent.LetS1=0,S2=0andS3=0betheequationsofthreecircleswithunitcoefficientsforx2andy2terms.ThentheradicalaxesofthecirclestakentwobytwoareS1−S2=0,S2−S3=0andS3−S1=0.
∴(S1−S2)+(S2−S3)+(S3−S1)≡0
Sincesumofthetermsvanishesidentically,thelinesrepresentedbyS1−S2=0,S2−S3=0andS3−S1=0areconcurrent.Thecommonpointofthelinesiscalledtheradicalcentre.
5.2ORTHOGONALCIRCLES
Definition5.2.1:Twocirclesaredefinedtobeorthogonalifthetangentsattheirpointofintersectionareatrightangles.
FindtheconditionforthecirclesS≡x2+y2+2gx+2fy+c=0,S1≡x2+y2
+2g1x+2f1y+c1=0tobeorthogonal.
LetPbeapointofintersectionofthetwocirclesS=0andS1=0.ThecentresareA(−g,−f),B(−g1,−f1).
Theradiiare
Sincethetwocirclesareorthogonal,PAisperpendiculartoPB.(i.e.)APBisarighttriangle.
Showthatifacirclecutstwogivencirclesorthogonallythenitscentreliesontheradicalaxisofthetwogivencircles.LetS1=x2+y2+2g1x+2f1y+c1=0andS2=x2+y2+2g2x+2f2y+c1=0bethetwogivencircles.LetS=x2+y2+2gx+2fy+c=0cutsS1=0andS2=0orthogonally.
SinceS=0cutsS1=0andS2=0orthogonally,
Bysubtracting,weget
Thisshowsthat(−g,−f)liesontheline,2(g1−g2)x+2(f1−f2)y+(c1−c2)=0whichistheradicalaxisofthetwocircles.Therefore,thecentreofthecircleS=0liesontheradicalaxisofthecirclesS1
=0andS2=0.
5.3COAXALSYSTEM
Definition5.3.1:Asystemofcirclesissaidtobecoaxalifeverypairofthesystemhasthesameradicalaxis.
Expresstheequationofacoaxalsystemofcirclesinthesimplestform.Inacoaxalsystemofcircles,everypairofthesystemhasthesameradicalaxis.Therefore,thereisacommonradicalaxistoacoaxalsystemofcircles.Hence,inacoaxalsystemthecentresareallcollinearandthecommonradical
axisisperpendiculartothelinesofcentres.Therefore,letuschoosethelineofcentresasx-axisandthecommonradicalaxisasy-axis.Letusconsidertwocirclesofthecoaxalsystem,
Sincethecentreslieonthex-axis,f1=0andf2=0.Therefore,theequationsofthecirclesarex2+y2+2gx+c=0andx2+y2+
2g1x+c1=0.Theradicalaxisofthesetwocirclesis2(g−g1)x+(c−c1)=0.
However,thecommonradicalaxisisthey-axiswhoseequationisx=0.
∴c−c1=0orc=c1.Hence,thegeneralequationtoacoaxalsystemofcirclesisx2+y2+2gx+c=0wheregisavariableandcisaconstant.
Sotheequationofacoaxalsystemcanbeexpressedinthesimplestform
x2+y2+2λx+c=0whereλisavariableandcisaconstant.
5.4LIMITINGPOINTS
Definition5.4.1:Limitingpointsaredefinedtobethecentresofpointcirclesbelongingtoacoaxalsystem;thatis,theyarecentresofcirclesofzeroradiibelongingtoacoaxalsystem.
Obtainthelimitingpointsofthecoaxalsystemofcirclesx2+y2+2λx+c=
0.Centresare(−λ,0)andradiiare
Forpointcirclesradiiarezero.
Therefore,limitingpointsare
Theorem5.4.1:Thepolarofonelimitingpointofacoaxalsystemofcircleswithrespecttoanycircleofthesystempassesthroughtheotherlimitingpoint.
Proof:Letx-axisbethelineofcentresandy-axisbethecommonradicalaxisofacoaxalsystemofcircles.Thenanycircleofthecoaxalsystemis
whereλisavariableandcisaconstant.
Thelimitingpointsofthiscoaxalsystemofcirclesare
Thepolarofthepoint withrespecttothecircle(5.1)is
Thislinepassesthroughtheotherlimitingpoint.Foreverycoaxalsystemofcirclesthereexistsanorthogonalsystemofcircles.Letx-axisbethelineofcentresandy-axisbethecommonradicalaxis.Thentheequationtoacoaxalsystemofcirclesis
Letusassumethatthecircle
cuteverycircleofthecoaxalsystemofcirclesgivenby(5.2)orthogonally.Thentheconditionfororthogonalityis
Letusnowconsidertwocirclesofthecoaxalsystemforthedifferentvaluesofλ,sayλ1andλ2.Thecondition(5.4)becomes2gλ1=c+k,2gλ2=c+k.
∴2(λ1−λ2)g=0.Sinceλ1−λ2≠0,g=0andsok=−c.Hence,from(5.3)theequationofthecirclewhichcutseverymemberofthe
system(5.2)isx2+y2+2fy−c=0,wherefisanarbitraryconstant.Therefore,foreverycoaxalsystemofcirclesthereexistsanorthogonalsystemofcirclesgivenbyx2+y2+2fy−c=0;wherefisavariableandcisaconstant.Forthis
systemoforthogonalcirclesy-axisisthelineofcentresandx-axisisthecommonradicalaxis.
Note5.4.1:Everycircleoftheorthogonalcoaxalsystemofcirclespasses
throughthelimitingpoints .
Theorem5.4.2:IfS=x2+y2+2gx+2fy+c=0andS1=x2+y2+2g1x+2f1y+c1=0beanytwocirclesofacoaxalsystemthenanycircleofcoaxalsystemcanbeexpressedintheformS+λS1=0.
Proof:
Consider,
whereλisavariable.Inthisequation,(1+λ)isthecoefficientofx2andy2.
Dividingby(1+λ)equation(5.7)becomes inwhichthecoefficientof
x2andy2areunity.
Nowconsidertwodifferentvaluesofλ,thatis,λ1andλ2.Then, and
Theradicalaxisofthesetwocirclesis
Sinceλ1−λ2≠0andthereforeS−S1=0whichisthecommonradicalaxis.Therefore,everymemberofthecoaxalsystemcanbeexpressedintheformS+λS1=0whereλisavariable.
Theorem5.4.3:IfS=x2+y2+2gx+2fy+c=0isacircleofacoaxalsystemandL=lx+my+n=0isthecommonradicalaxisofthesystemthenS+λL=0istheequationofacircleofthecoaxalsystemofcircles.
Proof:
Considertwomembersofthesystem(5.10)forthedifferentvaluesofλ,thatis,λ1andλ2.Then,S+λ1L=0andS+λ2L=0Theradicalaxisofthesetwocirclesis(λ1−λ2)L=0.
Sinceλ1−λ2≠0,L=0whichisthecommonradicalaxis.Therefore,S+λL=0representsanycircleofthecoaxalsysteminwhichS=0isacircleandL=0isthecommonradicalaxis.
5.5EXAMPLES(RADICALAXIS)
Example5.5.1
Findtheradicalaxisofthetwocirclesx2+y2+2x+4y−7=0andx2+y2−6x+2y−5=0andshowthatitisatrightanglestothelineofcentresofthetwocircles.
Solution
TheradicalaxisofthecirclesisS−S1=0.
Theslopeoftheradicalaxisism1=−4.Thecentresofthetwocirclesare(−1,−2)and(3,−1).
Theslopeofthelineofcentresis
Therefore,theradicalaxisisperpendiculartothelineofcentres.
Example5.5.2
Showthatthecirclex2+y2+2gx+2fy+c=0willbisectthecircumferenceofthecirclex2+y2+2g1x+2f1y+c1=0,if2g1(g−g1)+2f1(f−f1)=c−c1.
Solution
Let
Theradicalaxisofthesetwocirclesis2(g−g1)c+2(f−f1)y+c−c1=0,Circle(5.14)bisectsthecircumferenceofthecircle(515).Therefore,radicalaxispassesthroughthecentreofthesecondcircle.Theradicalaxisofthetwogivencirclesbe
Example5.5.3
Showthatthecirclesx2+y2−4x+6y+8=0andx2+y2−10x−6y+14=0toucheachotherandfindthecoordinatesofthepointofcontact.
Solution
Theradicalaxisofthesetwocirclesis6x+12y−6=0.
ThecentresofthecirclesareA(2,−3)andB(5,3).
Theradiiofthecirclesare
TheperpendiculardistancefromA(2,−3)ontheradicalaxisx+2y−1=0is
radiusofthefirstcircle.
Therefore,radicalaxistouchesthefirstcircleandhencethetwocirclestoucheachother.
Theequationofthelinesofcentresis
or
Solving(5.18)and(5.19),wegetthepointofcontact.Therefore,thepointofcontactis(3,−1).
Example5.5.4
Showthatthecirclesx2+y2+2ax+c=0andx2+y2+2by+c=0touchif
Solution
Theradicalaxisofthetwogivencirclesis2ax−2by=0.Thecentreofthefirst
circleis(−a,0).Theradiusofthefirstcircleis
Ifthetwocirclestoucheachother,thentheperpendiculardistancefromthecentre(−a,0)totheradicalaxisisequaltotheradiusofthecircle.
Ondividingbya2b2c,weget
Example5.5.5
Findtheradicalcentreofthecirclesx2+y2+4x+7=0,2x2+2y2+3x+5y+9=0andx2+y2+y=0.
Solution
Let
Theradicalaxisofcircles(5.20)and(5.22)is
Theradicalaxisofthecircles(5.20)and(5.22)is
Solving(5.23)and(5.24)wegettheradicalcentreasfollows:
Therefore,theradicalcentreis(−2,−1).
Example5.5.6
Provethatifthepointsofintersectionofthecirclesx2+y2+ax+by+c=0andx2+y2+a1x+b1y+c1=0bythelinesAx+By+C=0andA1x+B1y+C1=0areconcyclicif
Solution
Let
Ax+By+C=0meetsthecircle(5.25)atPandQandA1x+B1y+C1=0meetsthecircle(5.26)atRandS.SinceP,Q,RandSareconcyclic,theequationofthiscirclebe
Theradicalaxisofthecircles(5.25)and(5.26)is
Theradicalaxisofcircles(5.25)and(5.29)is
Theradicalaxisofcircles(5.26)and(5.29)is
Sincethesethreeradicalaxesareconcurrentwegetfromequations(5.30),(5.31)and(5.32),
Example5.5.7
Provethatthedifferenceofthesquareofthetangentstotwocirclesfromanypointintheirplanevariesasthedistanceofthepointfromtheirradicalaxis.
Solution
LetP(x1,y1)beanypointandthetwocirclesbe
Theequationtotheradicalaxisofthesetwocirclesbe
Theperpendiculardistanceofthepointfromtheradicalaxisis
Fromequations(5.36)and(5.37),weget
Example5.5.8
Provethatforallconstantsλandμ,thecircle(x−a)(x−a+λ)+(y−b)(y−b+μ)=r2bisectsthecircumferenceofthecircle(x−a)2+(y−b)2=r2.
Solution
Theradicalaxisofthesetwocirclesis
Thecentreofthesecondcircleis(a,b).Substitutingx=a,y=bin(5.40),weget
λa+μb−λa−μb=0.
∴(a,b)liesontheradicalaxis.Therefore,theradicalaxisbisectsthecircumferenceofthesecondcircle.
Example5.5.9
Provethatthelengthofcommonchordofthetwocirclesx2+y2+2λx+c=0
and
Solution
Thetwogivencirclesare
CentresareA(−λ,0)andB(−μ,0),radiiare
Theradicalaxisisλx−μy+c=0.TheperpendiculardistancefromAon
Therefore,thelengthofcommonchord
Example5.5.10
Showthatthecirclex2+y2−8x−6y+21=0isorthogonaltothecirclex2+y2
−2y−15=0.Findthecommonchordandtheequationofthecirclepassingthroughthecentresandintersectingpointsofthecircles.
Solution
Theconditionfororthogonalityis2gg1+2ff1=c+c1.
(i.e.)2(−4)(0)+2(−3)(−1)=21−150+6=6whichistrue.
Therefore,thetwocirclescuteachotherorthogonally.TheequationofthecommonchordisS−S1=0.
AnycirclepassingthroughtheintersectionofthecirclesisS+λL=0.
(i.e.)x2+y2−8x−6y+21+λ(2x+y−9)=0.
Thispassesthroughthecentre(4,3)ofthefirstcircle.
Therefore,theequationoftherequiredcircleisx2+y2−8x−6y+21+2(2x+y−9)=0.
(i.e.)x2+y2−4x−4y+3=0
Example5.5.11
Findtheequationtothecirclewhichcutsorthogonallythethreecirclesx2+y2+2x+17y+4=0,x2+y2+7x+6y+11=0andx2+y2−x+22f+33=0.
Solution
Lettheequationofthecirclewhichcutsorthogonallythethreegivencirclesbex2+y2+2gx+2fy+c=0.Thentheconditionsfororthogonalityare
From(5.45),weget3g+10=1
From(5.41),weget−6−34=c+4orc=−44Therefore,theequationofthecirclewhichcutsorthogonallythethreegiven
circlesisx2+y2−6x−4y−44=0.
Aliter
Theradicalaxisofcircles(5.41)and(5.42)is
Theradicalaxisofcircles(5.41)and(5.43)is
Therefore,radicalcentreis(3,2).
Therefore,radicalcentreis(3,2).IfRisthelengthofthetangentfrompoints(3,2)tothefirstcirclethenR2=9
+4+6+34+4=57.Therefore,theequationoftherequiredcircleis(x−3)2+(y−2)2=57.
(i.e.)x2+y2−6x−4y−44=0
Example5.5.12
Findtheequationofthecirclewhichpassesthroughtheorigin,hasitscentreonthelinex+y=4andcutsorthogonallythecirclex2+y2−4x+2y+4=0.
Solution
Lettheequationoftherequiredcirclepassingthroughtheoriginbe
Thiscirclecutsorthogonallythecircle
Thecentreofthecircle(5.48)liesonx+y=4.
Adding,weget
Therefore,theequationoftherequiredcircleisx2+y2−4x−4y=0.
Example5.5.13
IftheequationofthecircleswithradiirandRareS=0andS1=0,respectively
thenshowthatthecircles willintersectorthogonally.
Solution
Withoutlossofgenerality,wecanassumethelineofcentresofthetwocirclesasx-axisandthedistancebetweenthecentresas2a.Thenthecentresofthetwocirclesare(a,0)and(−a,0).TheequationofthetwocirclesareS=(x−a)2+y2
−r2=0andS1=(x+a)2+y2−R2=0.
Consider
∴RS±rS1=0Clearly,thecoefficientsoftheRandrintheseequationsarethesameandsotheyrepresentcircles.
ConsiderRS+rS1=0
Also,RS−rS=0hastheequation
Equations(5.52)and(5.53)canbewrittenas R=0and
Theconditionfororthogonalityis2gg1+2ff1=c+c1.
Therefore,thecircles areorthogonal.
Exercises(RadicalAxis)
1. Findtheradicalaxisofthecirclesx2+y2+2x+4y=0and2x2+2y2−7x−8y+1=0.Ans.:11x+16y−1=0
2. Findtheradicalaxisofthecirclesx2+y2−4x−2y−11=0andx2+y2−2x−6y+1=0andshowthattheradicalaxisisperpendiculartothelineofcentres.
Ans.:x−2y+5=0
3. Showthatthecirclesx2+y2−6x−9y+13=0andx2+y2−2x−16y=0toucheachother.Findthecoordinatesofpointofcontact.
Ans.:(5,1)
4. Findtheequationofthecommonchordofthecirclesx2+y2+2ax+2by+c=0andx2+y2+
2bx+2ay+c=0andalsoshowthatthecirclestouchif(a+b)2=2c.
5. Showthatthecirclesx2+y2+2x−8y+8=0andx2+y2+10x−2y+22=0toucheachotherandfindthepointofcontact.
Ans.:
6. Findtheequationofthecirclepassingthroughtheintersectionofthecirclesx2+y2=6andx2+
y2−6x+8=0andalsothroughthepoint(1,1).
Ans.:x2+y2−x−y=0
7. Findtheequationofthecirclepassingthroughthepointofintersectionofthecirclesx2+y2−6x
+2y+4=0andx2+y2+2x−4y−6=0andwhoseradiusis3/2.
Ans.:5x2+5y2−18x+y+5=0
8. Ifthecirclesx2+y2+2gx+2fy=0andx2+y2+2g1x+2f1y=0toucheachotherthenshowthatfg1=f1g.
9. Findtheradicalcentreofthecirclesx2+y2+aix+biy+c=0,i=1,2,3.Ans.:(0,0)
10. Findtheradicalcentreofthecirclesx2+y2−x+3y−3=0,x2+y2−2x+2y+2=0andx2+
y2+2x+2y−9=0.Ans.:(2,1)
11. Theradicalcentreofthreecirclesisattheorigin.Theequationoftwoofthecirclesarex2+y2=1
andx2+y2+4x+4y−1=0.Findthegeneralformofthethirdcircle.Ifitpassesthrough(1,1)and(−2,1)thenfinditsequation.
Ans.:x2+y2+x−2y−1=0
12. Findtheradicalcentreofthecirclesx2+y2+x+2y+3=0,x2+y2+4x+7=0and2x2+2y2+3x+5y+9=0.
Ans.:(−2,−1)
13. Findtheequationofthecirclewhoseradiusis3andwhichtouchesthecirclex2+y2−4x−6y+2=0internallyatthepoint(−1,−1).
14. Showthattheradicalcentresofthreecirclesdescribedonthesidesofatriangleasdiameteristheorthocentreofthetriangle.
15. Findtheequationofthecirclewhichcutsorthogonallythethreecirclesx2+y2+y=0,x2+4y2+
4x+7=0,21x2+y2+3x+5y+9=0.
Ans.:x2+y2+4x+2y+1=0
16. AandBaretwofixedpointsandPmovessothatPA=n·PB.ShowthatthelocusofPisacircleandthatfordifferentvaluesofn,allthecircleshavethesameradicalaxis.
17. Findtheequationofcirclewhoseradiusis5andwhichtouchesthecirclex2+y2−2x−4y−20=0atthepoint(5,5).
18. Provethatthelengthofthecommonchordofthetwocircleswhoseequationsare(x−a)2+(y−
b)2=r2and(x−a)2+(y−b)2=c2is
19. Findtheequationtotwoequalcircleswithcentres(2,3)and(5,6)whichcutseachotherorthogonally.
20. IfthreecircleswithcentresA,BandCcuteachotherorthogonallyinpairsthenprovethatthepolarofAwithrespecttothecirclecentreBpassesthroughC.
21. Findthelocusofcentresofallthecircleswhichtouchthelinex=2aandcutthecirclex2+y2=
a2orthogonally.22. A,Barethepoints(a,0)and(−a,0).ShowthatifavariablecircleSisorthogonaltothecircleon
ABasdiameter,thepolarof(a,0)withrespecttothecircleSpassesthroughthefixedpoint(−a,0).
23. Ifacirclepassesthroughthepoint(a,b)andcutsthecirclex2+y2=k2orthogonallythenprove
thatthelocusofitscentresis2ax+2by−(a2+b2+k2)=0.
24. Showthatthecirclesx2+y2+10x+6y+14=0andx2+y2−4x+6y+8=0toucheachotheratthepoint(3,−1).
25. Showthatthecirclesx2+y2+2ax+4ay−3a2=0andx2+y2−8ax−6ay+7a2=0toucheachotheratthepoint(a,0).
26. Theequationofthreecirclesarex2+y2=1,x2+y2+8x+15=0andx2+y2+10y+24=0.Determinethecoordinateofthepointsuchthatthetangentsdrawnfromittothethreecirclesareequalinlength.
27. IfPandQbeapairofconjugatepointswithrespecttoacircleS=0thenprovethatthecircleonPQasdiametercutsthecircleS=0orthogonally.
28. Findtheequationofthecirclewhosediameteristhecommonchordofthecirclesx2+y2+2x+
3y+1=0andx2+y2+4x+3y+2=0.
5.6EXAMPLES(LIMITINGPOINTS)
Example5.6.1
IfA,BandCarethecentresofthreecoaxalcirclesandt1,t2andt3arethelengthsoftangentstothemfromanypointthenprovethat
Solution
Letthethreecirclesofcoaxalsystembe
ThecentresareA(−g1,0),B(−g2,0)andC(−g3,0)andBC=g3−g2,CA=g1−g3,
Then,
sinceΣ(g2−g3)=0andΣg1(g2−g3)=0.
Example5.6.2
Findtheequationsofthecircleswhichpassthroughthepointsofintersectionofx2+y2−2x+1=0andx2+y2−5x−6y−4=0andwhichtouchtheline2x−y+3=0.
Solution
Theradicalaxisofthesetwocirclesis
Theequationofanycirclepassingthroughtheintersectionofthesetwocirclesisx2+y2−2x+1+λ(x+2y−1)=0.
Thecentreofthiscircleis andradius= Thecircle
(5.56)touchestheline2x−y+3=0.
Theequationofthecirclesarex2+y2−2x+1±2(x+2y−1)=0.(i.e.)x2+y2−2x+1+2x+4y−2=0andx2+y2−2x+1−2x−4y+2=0(i.e.)x2+y2+4y−1=0andx2+y2−4x+4y+3=0
Example5.6.3
Findtheequationofthecirclewhichpassesthroughtheintersectionofthe
circlesx2+y2=4andx2+y2−2x−4y+4=0andhasaradius
Solution
Theradicalaxisofthesetwocirclesis2x+4y−8=0.Anycirclepassingthroughtheintersectionofthesetwocirclesis
Centreis(−λ,−2λ).
Therefore,therequiredcirclesarex2+y2−4−2(2x+4y−8)=0and
(i.e.)x2+y2−4x−8y+12=0and5x2+5y2+4x+8y−36=0
Example5.6.4
Findtheequationofthecirclewhosediameteristhecommonchordofthecirclesx2+y2+2x+3y+1=0andx2+y2+4x+3y+2=0.
Solution
Theradicalaxisofthesetwocirclesis2x+1=0.Anycircleofthesystemisx2+y2+2x+3y+1+λ(2x+1)=0.
Centreis .
Sincetheradicalaxisisadiameter,centreliesontheradicalaxis.
Hence,theequationoftherequiredcircleis
Example5.6.5
Findtheequationofthecirclewhichtouchesx-axisandiscoaxalwiththecirclesx2+y2+12x+8y−33=0andx2+y2=5.
Solution
Theradicalaxisofthesetwocirclesis
Anycircleofthecoaxalsystemisx2+y2−5+λ(6x+4y−14)=0.Centreis(−3λ,−2λ).
Thecircles(5.69)touchesx-axis(i.e.)y=0.
Therefore,thetwocirclesofthesystemtouchingx-axisare
Example5.6.6
Theline2x+3y=1cutsthecirclex2+y2=4inAandB.ShowthattheequationofthecircleonABasdiameteris13(x2+y2)−4x−6y−50=0.
Solution
Let
Anycirclepassingthroughtheintersectionofthecircleandthelineis
Centreis andradius=
IfABisadiameterofthecircle(5.72),theircentreshouldlieonAB.
Therefore,theequationofthecircleonABasdiameteris13(x2+y2−4)−2(2x+3y−1)=0.
∴13(x2+y2)−4x−6y+50=0
Example5.6.7
Apointmovessothattheratioofthelengthoftangentstothecirclesx2+y2+4x+3=0andx2+y2−6x+5=0is2:3.Showthatthelocusofthepointisacirclecoaxalwiththegivencircles.
Solution
ThelengthsoftangentsfromapointP(x1,y1)tothetwocirclesare
Giventhat,
Thelocusof
ThisisoftheformS1+λS2=0Hencethelocusofcircleisacirclecoaxalwiththetwogivencircles.
Example5.6.8
Findthelimitingpointsofthecoaxalsystemdeterminedbythecirclex2+y2+2x+4y+7=0andx2+y2+4x+2y+5=0.
Solution
Giventhat,
Theradicalaxisofthesetwocirclesis2x−2y−2=0.Anycircleofthecoaxalsystemisx2+y2+2x+4y+7+λ(2x−2y−2)=0.Centreis(−1−λ,−2+λ).
Radiusis
Limitingpointsarethecentresofcirclesofradiizero.Therefore,limitingpointsare(−2,−1)and(0,−3).
Example5.6.9
Thepoint(2,1)isalimitingpointofasystemofcoaxalcirclesofwhichx2+y2
−6x−4y−3=0isamember.Findtheequationtotheradialaxisandthecoordinatesoftheotherlimitingpoint.
Solution
Giventhat
x2+y2−6x−4y−3=0Since(2,1)isalimitpoint,thepointcirclecorrespondingtothecoaxalsystemis
Theradicalaxisofthesystemis
AnycircleofthecoaxalsystemisS+λL=0.
x2+y2−6x−4y−3+λ(2x+2y+8)=0
Centreis(3−λ,2−λ).
Radius
Forpointcircles,radius=0.
Therefore,thelimitingpointsarethecentresofpointcircleofthecoaxalsystem,thatis,(2,1)and(−5,−6).
Example5.6.10
Findtheequationofthecirclewhichpassesthoughtheoriginandbelongstothecoaxalsystemofwhichlimitingpointsare(1,2)and(4,3).
Solution
Since(1,2)and(4,3)arelimitingpointsoftwocirclesofthecoaxalsystemand(x−1)2+(y−2)2=0and(x−4)2+(y−3)2=0.
Radicalaxisis6x+2y−20=0.Anycircleofthesystemisx2+y2−2x−4y+5+λ(6x+2y−20)=0.Thispassesthroughtheorigin.
Hence,theequationofthesystemis
Example5.6.11
ApointPmovessothatitsdistancesfromtwofixedpointsareinaconstantratioλ.ProvethatthelocusofPisacircle.IfλvariesthenshowthatPgeneratesasystemofcoaxalcirclesofwhichthefixedpointsarethelimitingpoints.
Solution
LetP(x1,y1)beamovingpointandA(c,0)andB(0,−c)bethetwofixedpoints.Here,wehavechosenthefixedpointsonthex-axissuchthatPisitsmidpoint.Giventhat
ThisequationisoftheformS+λS′=0whichistheequationtoacoaxalsystemofcircles.Therefore,fordifferentvaluesofλ,Pgeneratesacoaxalsystemofcirclesof
which(x−a)2+y2=0and(x+a)2+y2=0aremembers.Theseequationsaretheequationofpointcircleswhosecentresare(a,0),(−a,0)whichisthefixedpoints.
Example5.6.12
Provethatthelimitingpointofthesystemx2+y2+2gx+c+λ(x2+y2+2fy+k)
=0subtendsarightangleattheoriginif
Solution
Thetwomembersofthesystemarex2+y2+2gx+c=0andx2+y2+2fy+k=0.Radicalaxisis2gx−2fy+c−k=0.Anycircleofthesystemisx2+y2+2gx+c+λ(2gx−2fy+c−k)=0.Centreis(−g−gλ,fλ).
Radius
Forpointcircle,radius=0.
Consideringthetwovaluesofλasλ1,λ2,
centresareA(−g(1+λ1),fλ1)andB(−g(1+λ2),fλ2)SinceOABisrightangledatO,OAisperpendiculartoOB.
Exercises
1. Findtheequationofthecirclepassingthroughtheintersectionofx2+y2−6=0andx2+y2+4y−1=0throughthepoint(−1,1).
Ans.:9x2+9y2+16y−34=0
2. Showthatthecirclesx2+y2=480andx2+y2−10x−24y+120=0toucheachotherandfindtheequation,ifathirdcirclewhichtouchesthecirclesattheirpointofintersectionandthex-axis
x2+y2−200x−400y+10000=0.
Ans.:5x2+5y2−40x+96y+30=0
3. Findtheequationofthecirclewhosecentreliesonthelinex+y−11=0andwhichpasses
throughtheintersectionofthecirclex2+y2−3x+2y−4=0withtheline2x+5y−2=0.
4. Findthelengthofthecommonchordofthecirclesx2+y2+4x−22y=0andx2+y2−10x+5y=0.
Ans.:40/7
5. Findthecoordinatesofthelimitingpointsofthecoaxalcirclesdeterminedbythetwocirclesx2+
y2−4x−6y−3=0andx2+y2−24x−26y+277=0.Ans.:(1,2),(3,1)
6. Findthecoordinatesofthelimitingpointsofthecoaxalsystemofcirclesofwhichtwomembers
arex2+y2+2x−6y=0and2x2+2y2−10y+5=0.Ans.:(1,2),(3,1)
7. Findthecoaxalsystemofcirclesifoneofwhosemembersisx2+y2+2x−6y=0andalimitingpointis(1,−2).
Ans.:x2+y2+2x+3y−7−λ(4x−y−12)=0
8. Findthelimitingpointofthecoaxalsystemdeterminedbythecirclesx2+y2−6x−6y+4=0
andx2+y2−2x−4y+3=0.
Ans.:
9. Findtheequationofthecoaxalsystemofcirclesoneofwhosemembersisx2+y2−4x−2y−5=0andthelimitingpointis(1,2).
Ans.:x2+y2−2x−4y+5+λ(x−y−5)=0
10. Iforiginisalimitingpointofasystemofcoaxalcirclesofwhichx2+y2+2gx+2fy+c=0isa
memberthenshowthattheotherlimitingpointsis
11. Showthattheequationofthecoaxalsystemwhoselimitingpointsare(0,0)and(a,b)isx2+y2+
k(2ax−2by−a2−b2)=0.
12. Theoriginisalimitingpointofasystemofcoaxalcirclesofwhichx2+y2+2gx+2fy+c=0isa
member.Showthattheequationofcirclesoftheorthogonalsystemis(x2+y2)(g+λf)+c(x−λy)=0fordifferentvaluesofx.
13. Showthatthecirclesx2+y2+2ax+2by+2λ(ax−by)=0whereλisaparameterfromacoaxalsystemandalsoshowthattheequationofthecommonradicalaxisandtheequationofcircles
whichareorthogonaltothissystemare
14. ApointPmovessuchthatthelengthoftangentstothecirclesx2+y2−2x−4y+5=0andx2+
y2+4x+6y−7=0areintheratio3:4.Showthatthelocusisacircle.
15. Showthatthelimitingpointsofthecirclex2+y2=a2andanequalcirclewithcentreontheline
lx+my+n=0beontheline(x2+y2)(lx+my+n)+a2(ln+mn)=0.
Chapter6
Parabola
6.1INTRODUCTION
Ifapointmovesinaplanesuchthatitsdistancefromafixedpointbearsaconstantratiotoitsperpendiculardistancefromafixedstraightlinethenthepathdescribedbythemovingpointiscalledaconic.Inotherwords,ifSisafixedpoint,lisafixedstraightlineandPisamovingpointandPMisthe
perpendiculardistancefromPonl,suchthat constant,thenthelocusofP
iscalledaconic.Thisconstantiscalledtheeccentricityoftheconicandisdenotedbye.
Ife=1,theconiciscalledaparabola.Ife<1,theconiciscalledanellipse.Ife>1,theconiciscalledahyperbola.ThefixedpointSiscalledthefocusoftheconic.Thefixedstraightlineis
calledthedirectrixoftheconic.Theproperty iscalledthefocus-directrix
propertyoftheconic.
6.2GENERALEQUATIONOFACONIC
Wecanshowthattheequationofaconicisaseconddegreeequationinxandy.Thisisderivedfromthefocus-directrixpropertyofaconic.LetS(x1,y1)bethefocusandP(x,y)beanypointontheconicandlx+my+n=0betheequationofthedirectrix.Thefocus-directrixpropertyoftheconicstates
(i.e.)
Thisequationcanbeexpressedintheformax2+2hxy+by2+2gx+2fy+c=0whichisaseconddegreeequationinxandy.
6.3EQUATIONOFAPARABOLA
LetSbethefocusandthelinelbethedirectrix.WehavetofindthelocusofapointPsuchthatitsdistancefromthefocusSisequaltoitsdistancefromthefixedlinel.
(i.e.) wherePMisperpendiculartothedirectrix.
DrawSXperpendiculartothedirectrixandbisectSX.LetAbethepointofbisectionandSA=AX=a.ThenthepointAisapointontheparabolasince
.TakeASasthex-axisandAYperpendiculartoASasthey-axis.Thenthe
coordinateofSare(a,0).Let(x,y)bethecoordinatesofthepointP.DrawPNperpendiculartothex-axis.
This,beingthelocusofthepointP,istheequationoftheparabola.Thisequationisthesimplestpossibleequationtoaparabolaandiscalledthestandardequationoftheparabola.
Note6.3.1:
1. ThelineAS(x-axis)iscalledtheaxisoftheparabola.2. ThepointAiscalledthevertexoftheparabola.3. AY(y-axis)iscalledthetangentatthevertex.4. Theperpendicularthroughthefocusiscalledthelatusrectum.5. Thedoubleordinatethroughthefocusiscalledthelengthofthelatusrectum.6. Theequationofthedirectrixisx+a=0.7. Theequationofthelatusrectumisx–a=0.
6.4LENGTHOFLATUSRECTUM
Tofindthelengthofthelatusrectum,drawLM′perpendiculartothedirectrix.
Then
6.4.1Tracingofthecurvey2=4ax
1. Ifx<0,yisimaginary.Therefore,thecurvedoesnotpassthroughtheleftsideofy-axis.2. Wheny=0,wegetx=0.Therefore,thecurvemeetsthey-axisatonlyonepoint,thatis,(0,0).
3. Whenx=0,y2=0,thatis,y=0.Hencethey-axismeetsthecurveattwocoincidentpoints(0,0).Hencethey-axisisatangenttothecurveat(0,0).
4. If(x,y)isapointontheparabolay2=4ax,(x,–y)isalsoapoint.Therefore,thecurveis
symmetricalaboutthex-axis.5. Asxincreasesindefinitely,thevaluesofyalsoincreasesindefinitely.Thereforethepointsofthe
curvelyingontheoppositesidesofx-axisextendtoinfinitytowardsthepositivesideofx-axis.
6.5DIFFERENTFORMSOFPARABOLA
1. Ifthefocusistakenatthepoint(–a,0)withthevertexattheoriginanditsaxisasx-axisthenits
equationisy2=–4ax.
2. Iftheaxisoftheparabolaisthey-axis,vertexattheoriginandthefocusat(0,a),theequationof
theparabolaisx2=4ay.
3. Ifthefocusisat(0,−a),vertex(0,0)andaxisasy-axis,thentheequationoftheparabolaisx2=–4ay.
ILLUSTRATIVEEXAMPLESBASEDONFOCUSDIRECTRIXPROPERTY
Example6.1
Findtheequationoftheparabolawiththefollowingfocianddirectrices:i. (1,2):x+y–2=0ii. (1,–1):x–y=0iii. (0,0):x–2y+2=0
Solution
i. LetP(x,y)beanypointontheparabola.DrawPMperpendiculartothedirectrix.Thenfromthe
definitionoftheparabola,
∴SP2=(x–1)2+(y–2)2PM=perpendiculardistancefrom(x,y)onx+y–2=0
Thisistheequationoftherequiredparabola.ii. ThepointSis(1,−1).Directrixisx–y=0
Fromanypointontheparabola,
iii. Sis(0,0).Directrixisx–2y+2=0
ForanypointPontheparabola,
Example6.2
Findthefoci,latusrectum,verticesanddirectricesofthefollowingparabolas:i. y2+4x–2y+3=0
ii. y2–4x+2y–3=0
iii. y2–8x–9=0
Solution
i.
Takex+ =X,y–1=Y.Shiftingtheorigintothepoint theequationoftheparabola
becomesY2=−4X.
∴Vertexis ,latusrectumis4,focusis andfootofthedirectrixis .The
equationofthedirectrixisx= or2x–1=0.
ii.
Shiftingtheorigintothepoint(–1,–1)bytakingx+1=Xandy+1=Ytheequationofthe
parabolabecomesY2=4X.∴Vertexis(−1,−1),latusrectum=4,focusis(0,−1)andfootofthedirectrixis(−2,−1).∴Theequationofthedirectrixisx+2=0.
y2–8x–9=0⇒y2=8x+9
Shifttheorigintothepoint andtake
∴TheequationoftheparabolabecomesY2=8X.Vertexis ,latusrectum=8andfocusis
Theequationofthedirectrixis
Exercises
1. Findtheequationoftheparabolawhosefocusis(2,1)anddirectrixis2x+y+1=0.
Ans.:x2–4xy+4y2–24x–12y+24=0
2. Findtheequationoftheparabolawhosefocusis(3,−4)andwhosedirectrixisx–y+5=0.
Ans.:x2+2xy+y2–16x–26y+25=0
3. Findthecoordinatesofthevertex,focusandtheequationofthedirectrixoftheparabola3y2=16x.Findalsothelengthofthelatusrectum.
Ans.:
4. Findthecoordinatesofthevertexandfocusoftheparabola2y2+3y+4x=2.Findalsothelengthofthelatusrectum.
Ans.:
5. Apointmovesinsuchawaythatthedistancefromthepoint(2,3)isequaltothedistancefromtheline4x+3y=5.Findtheequationofitspath.Whatisthenameofthiscurve?
Ans.:25[(x–2)2+(y–3)2]–(4x+3y–5)2
6.6CONDITIONFORTANGENCY
Findtheconditionforthestraightliney=mx+ctobeatangenttotheparabolay2=4axandfindthepointofcontact.
Solution
Theequationoftheparabolais
Theequationofthelineis
Solvingequations(6.1)and(6.2),wegettheirpointsofintersection.Thex-coordinatesofthepointsofintersectionaregivenby
Ify=mx+cisatangenttotheparabola,thentherootsofthisequationareequal.Theconditionforthisisthediscriminantisequaltozero.
Hence,theconditionfory=mx+ctobeatangenttotheparabolay2=4axisc=a/m.Substitutingc=a/minequation(6.3),weget
Therefore,thepointofcontactis
Note6.6.1:Anytangenttotheparabolais
6.7NUMBEROFTANGENTS
Showthattwotangentscanalwaysbedrawnfromapointtoaparabola.
Solution
Lettheequationtotheparabolabey2=4ax.Let(x1,y1)bethegivenpoint.Any
tangenttotheparabolais Ifthistangentpassesthrough(x1,y1),then
(1)Thisisaquadraticequationinm.
Therefore,therearetwovaluesofmandforeachvalueofmthereisatangent.Hence,therearetwotangentsfromagivenpointtotheparabola.
Note6.7.1:Ifm1,m2aretheslopesofthetwotangentsthentheyaretherootsofequation(6.3).
6.8PERPENDICULARTANGENTS
Showthatthelocusofthepointofintersectionofperpendiculartangentstoaparabolaisthedirectrix.
Solution
Lettheequationoftheparabolabey2=4ax.Let(x1,y1)bethepointofintersectionofthetwotangentstotheparabola.Anytangenttotheparabolais
Ifthistangentpassesthrough(x1,y1)then
Ifm1,m2aretheslopesofthetwotangentsfrom(x1,y1),thentheyaretherootsofequation(6.5).Sincethetangentsareperpendicular,
Therefore,thelocusof(x1,y1)isx+a=0,whichisthedirectrix.Showthatthelocusofthepointofintersectionoftwotangentstotheparabola
thatmakecomplementaryangleswiththeaxisisalinethroughthefocus.
Solution
Let(x1,y1)bethepointofintersectionoftangentstotheparabolay2=4ax.Any
tangenttotheparabolais Ifthislinepassesthrough(x1,y1),then
Ifm1,m2aretheslopesofthetwotangents,then
Ifthetangentsmakecomplementaryangleswiththeaxisoftheparabola,thenm1=tanθandm2=tan(90–θ).
Thelocusofthepointofintersectionofthetangentsisx–a=0,whichisastraightlinethroughtheorigin.
6.9EQUATIONOFTANGENT
Findtheequationofthetangentat(x1,y1)totheparabolay2=4ax.LetP(x1,y1)andQ(x2,y2)betwopointsontheparabolay2=4ax.Then
Theequationofthechordjoiningthepoints(x1,y1)and(x2,y2)is
Fromequations(6.6)and(6.7),weget
Hence,theequationofthechordPQis
WhenthepointQ(x2,y2)tendstocoincidewithP(x1,y1),thechordPQbecomesthetangentatP.Hence,theequationofthetangentatPis
Aliter:Theequationoftheparabolaisy2=4ax.
Differentiatingthisequationwithrespecttox1,weget
Theequationofthetangentat(x1,y1)is
6.10EQUATIONOFNORMAL
Findtheequationofthenormalat(x1,y1)ontheparabolay2=4ax.
Solution
Theslopeofthetangentat(x1,y1)is
Therefore,theslopeofthenormalat(x1,y1)is
Theequationofthenormalat(x1,y1)is
6.11EQUATIONOFCHORDOFCONTACT
Findtheequationofthechordofcontactoftangentsfrom(x1,y1)totheparabolay2=4ax.
Solution
LetQRbethechordofcontactoftangentsfromP(x1,y1).LetQandRbethepoints(x2,y2)and(x3,y3),respectively.Then,theequationoftangentsatQandRare
ThesetwotangentspassthroughP(x1,y1).
Thesetwoequationsshowthatthepoints(x2,y2)and(x3,y3)lieonthelineyy1=2a(x+x1).Therefore,theequationofthechordofcontactoftangentsfromP(x1,y1)is
yy1=2a(x+x1).
6.12POLAROFAPOINT
Findthepolarofthepointwithrespecttotheparabolay2=4ax.
Definition6.12.1Thepolarofapointwithrespecttoaparabolaisdefinedasthelocusofthepointofintersectionofthetangentsattheextremitiesofachordpassingthroughthatpoint.
Solution
LetP(x1,y1)bethegivenpoint.LetQRbeavariablechordpassingthroughP.LetthetangentsatQandRintersectat(h,k).Thentheequationofthechordofcontactoftangentsfrom(h,k)isyk=2a(x+h).ThischordpassesthroughP(x1,y1).
∴y1k=2a(x1+h)Thenthelocusof(h,k)isyy1=2a(x+x1)Hence,thepolarof(x1,y1)withrespecttoy2=4axis
yy1=2a(x+x1)
Note6.12.1:PointPisthepoleofthelineyy1=2a(x+x1).
Note6.12.2:Findthepoleofthelinelx+my+n=0withrespecttotheparabolay2=4ax.Let(x1,y1)bethepole.Thenthepolarof(x1,y1)is
Butthepolarof(x1,y1)isgivenbylx+my+n=0(6.13)
Equations(6.12)and(6.13)representthesameline.Then,identifyingthesetwoequations,weget
Hence,thepoleofthelineis
6.13CONJUGATELINES
Definition6.13.1Twolinesaresaidtobeconjugatetoeachotherifthepoleofeachliesontheother.Findtheconditionforthelineslx+my+n=0andl1x+m1y+n1=0tobeconjugatelineswithrespecttotheparabolay2=4ax.
Solution
Let(x1,y1)bethepoleofthelinesl1x+m1y+n1=0withrespecttotheparabola.Thepolarof(x1,y1)withrespecttothepolary2=4axislx+my+n=0.Theequationofthepolarof(x1,y1)withrespecttotheparabolay2=4axis
Butthepolarof(x1,y1)isgivenbylx+my+n=0(6.15)
Equations(6.14)and(6.15)representthesameline.Identifyingthesetwoequations,weget
Thepoleofthelinelx+my+n=0is
Sincethelineslx+my+n=0andl1x+m1y+n1=0areconjugatetoeachother,thepoleoflx+my+n=0willlieonl1x+m1y+n1=0.
Thisistherequiredcondition.
6.14PAIROFTANGENTS
Findtheequationofpairoftangentsfrom(x1,y1)totheparabolay2=4ax.
Solution
Theequationofalinethrough(x1,y1)is
Anypointonthislineis(x1+rcosθ,y1+rsinθ).Thepointsofintersectionofthelineandtheparabolaaregivenby
Thetwovaluesofrofthisequationarethedistancesofpoint(x,y)tothepoint(x1,y1).Ifline(6.16)isatangenttotheparabola,thenthetwovaluesofrmustbeequalandtheconditionforthisisthediscriminantofquadratic(6.17)iszero.
∴4(y1sinθ–2acosθ)2=4sin2θ(y2–4ax1)Eliminatingθinthisequationwiththehelpof(6.16),weget
Therefore,theequationofpairoftangentsfrom(x1,y1)is
6.15CHORDINTERMSOFMIDPOINT
Findtheequationofachordoftheparabolaintermsofitsmiddlepoint(x1,y1).
Solution
Lettheequationofthechordbe
Anypointonthislineis(x1+rcosθ,y1+rsinθ).Whenthechordmeetstheparabolay2=4ax,thispointliesonthecurve.
ThetwovaluesofrarethedistancesRPandRQ,whichareequalinmagnitudebutoppositeinsign.Theconditionforthisisthecoefficientofrisequaltozero.
Thisistherequiredequationofthechord.
6.16PARAMETRICREPRESENTATION
x=at2,y=2atsatisfytheequationy2=4ax.Thismeans(at2,2at)isapointontheparabola.Thispointisdenotedby‘t’andtiscalledaparameter.
6.17CHORDJOININGTWOPOINTS
Findtheequationofthechordjoiningthepoints onthe
parabolay2=4ax.
Solution
Theequationofthechordjoiningthepointsis
Note6.17.1:Thechordbecomesthetangentat‘t’ift1=t2=t.Therefore,theequationofthetangentattis
y(2t)=2x+2at2oryt=x+at2
6.18EQUATIONSOFTANGENTANDNORMAL
Findtheequationofthetangentandnormalat‘t’ontheparabolay2=4ax.
Solution
Theequationoftheparabolaisy2=4ax.Differentiatingwithrespecttox,
Theequationofthetangentattis
Theslopeofthenormalattis−t.Theequationofthenormalat‘t’is
6.19POINTOFINTERSECTIONOFTANGENTS
Findthepointofintersectionoftangentsatt1andt2ontheparabolay2=4ax.
Solution
Theequationoftangentsatt1andt2are
Hence,thepointofintersectionis[at1t2,a(t1+t2)].
6.20POINTOFINTERSECTIONOFNORMALS
Findthepointofintersectionofnormalsatt1andt2.
Solution
6.21NUMBEROFNORMALSFROMAPOINT
Showthatthreenormalscanalwaysbedrawnfromagivenpointtoaparabola.
Solution
Lettheequationoftheparabolabey2=4ax.Theequationofthenormalattis
y+xt=2at+at3Ifthispassesthrough(x1,y1)then
Thisbeingacubicequationint,therearethreevaluesfort.Foreachvalueoftthereisanormalfrom(x1,y1)totheparabolay2=4ax.
Note6.21.1:Ift1,t2,t3aretherootsofequation(6.18),then
Note6.21.2:From(6.18),2at1+2at2+2at3=0Therefore,thesumofthecoordinatesofthefeetofthenormalisalwayszero.
6.22INTERSECTIONOFAPARABOLAANDACIRCLE
Provethatacircleandaparabolameetatfourpointsandshowthatthesumoftheordinatesofthefourpointsofintersectioniszero.
Solution
Lettheequationofthecirclebe
Lettheequationoftheparabolabe
Lettheequationoftheparabolabe
Anypointontheparabolais(at2,2at).Whenthecircleandtheparabolaintersect,thispointliesonthecircle,
Thisbeingafourthdegreeequationint,therearefourvaluesoft.Foreachvalueoftthereisapointofintersection.Hence,therearefourpointsofintersectionofacircleandaparabola.Ift1,t2,t3,t4bethefourrootsofequation(6.24),then
Multiplyingequation(6.25)by2a,weget
2at1+2at2+2at3+2at4=0Therefore,thesumoftheordinatesofthefourpointsofintersectioniszero.
ILLUSTRATIVEEXAMPLESBASEDONTANGENTSANDNORMALS
Example6.3
Findtheequationsofthetangentandnormaltotheparabolay2=4(x–1)at(5,4).
Solution
y2=4(x–1)Differentiatingwithrespecttox,
∴Theequationofthetangentat(5,4)isy–4= (x–5).
2y–8=x–5orx–2y+3=0.Theslopeofthenormalat(5,4)is−2.∴Theequationofnormalat(5,4)isy–4=−2(x–5)or2x+y=14.
Example6.4
Findtheconditionthatthestraightlinelx+my+n=0isatangenttotheparabola.
Solution
Anystraightlinetangenttotheparabolay2=4axisoftheformy=mx+cif
Considerthelinelx+my+n=0(i.e.)my=−lx−n
Ifthisisatangenttotheparabola,y2=4axthen
Example6.5
Acommontangentisdrawntothecirclex2+y2=r2andtheparabolay2=4ax.Showthattheangleθwhichitmakeswiththeaxisoftheparabolaisgivenby
Solution
Lety=mx+cbeacommontangenttotheparabola
andthecircle
Ify=mx+cistangenttotheparabola(6.27)then
Ify=mx+cisatangenttothecircle(6.29)then
Equations(6.29)and(6.30)representthesamestraightline.Identifyingweget
Sincem2hastobepositive,
Example6.6
Astraightlinetouchesthecirclex2+y2=2a2andtheparabolay2=8ax.Showthatitsequationisy=±(x+2a).
Solution
Theequationofthecircleis
Theequationoftheparabolais
Atangenttotheparabola(6.32)is
Atangenttothecircle(6.33)is
Equations(6.33)and(6.34)representthesamestraightline.Identifyingweget,
m2=1or−2;m2=−2isimpossible.
∴m2=1orm=±1∴Theequationofthecommontangentisy=±x±2a.
∴y=±(x+2a)
Example6.7
Showthatforallvaluesofm,theliney=m(x+a)+ willtouchtheparabolay2
=4a(x+a).Henceshowthatthelocusofapoint,thetwotangentsformwhichtotheparabolasy2=4a(x+a)andy2=4b(x+b)onetoeachareatrightangles,isthelinex+a+b=0.
Solution
Solving(6.35)and(6.36),wegettheirpointsofintersection.Thex-coordinatesoftheirpointsofintersectionaregivenby,
∴Thetwovaluesofxandhenceofyofthepointsofintersectionarethesame.
Hence, isatangenttotheparabolay2=4a(x+a).Let(x1,y1)be
thepointofintersectionofthetwotangentstotheparabolay2=4a(x+a),y2=
4a(y+b).Thetangentsare and Sincetheypass
through(x1,y1),wehave
Sincethetangentsareatrightangles,m1m2=−1.Subtracting(6.38)from(6.37),weget
since
Cancelling
Thelocusof(x1,y1)isx+a+b=0.
Example6.8
Provethatthelocusofthepointofintersectionoftwotangentstotheparabolay2
=4ax,whichmakesanangleofαwithx-axis,isy2–4ax=(x+a)2tan2α.Determinethelocusofpointofintersectionofperpendiculartangents.
Solution
Let(x1,y1)bethepointofintersectionoftangents.Anytangenttotheparabola
isy=mx+ .Ifthispassesthrough(x1,y1)theny=mx1+ .
∴Thelocusof(x1,y1)isy2–4ax=(x+a)2tan2α.Ifthetangentsareperpendicular,tanα=tan90°=∞∴Thelocusofperpendiculartangentsisdirectrix.
Example6.9
Provethatiftwotangentstoaparabolaintersectonthelatusrectumproducedthentheyareinclinedtotheaxisoftheparabolaatcomplementaryangles.
Solution
Let(x1,y1)theequationoftheparabolabey2=4ax.Lety=mx+ beany
tangenttotheparabola.Letthetwotangentsintersectat(a,y1),apointonthe
latusrectum.Then(a1,y1)lies,ony=mx+ .
Ifm1andm2aretheslopesofthetwotangentstotheparabolathenm1m2=1.(i.e.)tanθ·tan(90–θ)=1.(i.e.)Thetangentmakescomplementaryanglestotheaxisoftheparabola.
Example6.10
Provethatthelocusofpolesofthechordsoftheparabolay2=4axwhichsubtendsaconstantangleαatthevertexisthecurve(x+4a)2tan2α=4(y2–4ax).
Solution
Let(x1,y1)bethepoleofachordoftheparabola.Thenthepolarof(x1,y1)is
whichisthechordofcontactfrom(x1,y1).ThecombinedequationofthelinesAQandARisgotbyhomogenizationoftheequationoftheparabolay2=4axwiththehelpof(6.40).∴Thecombinedequationofthelinesis
Thelocusof(x1,y1)is(x+4a)2tan2α=4(y2–4ax).
Note6.10.1:Ifα=90°,thelocusof(x1,y1)isx+4a=0.
Example6.11
Iftwotangentsaredrawntoaparabolamakingcomplementaryangleswiththeaxisoftheparabola,provethatthechordofcontactpassesthroughthepointwheretheaxiscutsthedirectrix.
Solution
Lety=mx+ beatangentfromapoint(x1,y1)totheparabolay2=4ax.Then
y1=mx1+ orm2x1–my1+a=0.Ifthetangentsmakecomplementaryangles
withtheaxisoftheparabolathenm1m2=1or
Theequationofthechordofcontactfrom(a,y1)totheparabolaisyy1=2a(x+a).Whenthechordofcontactmeetsthex-axis,y=0.
∴x+a=0orx=−a.∴Thechordofcontactpassesthroughthepoint(−a,0)wheretheaxiscutsthedirectrix.
Example6.12
Findthelocusofpolesoftangentstotheparabolay2=4axwithrespecttotheparabolax2=4by.
Solution
Let(x1,y1)bethepolewithrespecttotheparabolax2=4by.Thenthepolarof
(x1,y1)isxx1=2b(y+y1), .Thisisatangenttotheparabolay2=4ax.
Theconditionfortangentisc= (i.e.) orx1y1+2ab=0.
Thelocusof(x1,y1)isthestraightlinexy+2ab=0.
Example6.13
Fromavariablepointonthetangentatthevertexofaparabola,theperpendicularisdrawntoitspolar.Showthattheperpendicularpassesthroughafixedpointontheaxisoftheparabola.
Solution
Theequationofthetangentatthevertexisx=0.Anypointonthislineis(0,y1).Thepolarof(0,y1)withrespecttotheparabolay2=4axisyy1=2ax.Theequationoftheperpendiculartothepolarof(x1,y1)isy1x+2ay=k.This
passesthrough(0,y1).
∴k=2ay1.∴Theequationoftheperpendiculartothepolarfrom(0,y1)isy1x+2ay=2ay1,whenthislinemeetsthex-axis,y1x=2ay1orx=2aHence,theperpendicularpassesthroughthepoint(2a,0),afixedpointonthe
axisoftheparabola.
Example6.14
Thepolarofanypointwithrespecttothecirclex2+y2=a2touchestheparabolay2=4ax.Showthatthepointliesontheparabolay2=–ax.
Solution
Thepolarofthepoint(x1,y1)withrespecttothecirclex2+y2=a2isxx1+yy1=a2
Thisisatangenttotheparabolay2=4ax.Theconditionforthatisc= .
Thelocusof(x1,y1)isy2=–axwhichisaparabola.
Example6.15
Findthelocusofpoleswithrespecttotheparabolay2=4axoftangentstothecirclex2+y2=c2.
Solution
Letthepolewithrespecttotheparabolay2=4axbe(x1,y1).Thenthepolarof(x1,y1)is
yy1=2a(x+x1)
(i.e.) .Thisisatangenttothecirclex2+y2=a2.Theconditionfor
thisis‘c2=a2(1+m2).
Thelocusof(x1,y1)is4a2x2=c2(y2+4a2).
Example6.16
ApointPmovessuchthatthelinethroughitperpendiculartoitspolarwithrespecttotheparabolay2=4axtouchestheparabolax2=4by.ShowthatthelocusofPis2ax+by+4a2=0.
Solution
LetPbethepoint(x1,y1).ThepolarofPwithrespecttoy2=4axis
Theequationoftheperpendicularto(6.41)isy1x+2ay+k=0.Thispassesthrough(x1,y1).
Hence,theequationoftheperpendicularisy1x+2ay–(x1y1+2ay1)=0.
(i.e.)
Thisisatangenttotheparabolax2=4by.
∴Thecondition
∴Thelocusof(x1,y1)is2ax+by+4a2=0.
Example6.17
IfthepolarofthepointPwithrespecttoaparabolapassesthroughQthenshowthatthepolarofQpassesthroughP.
Solution
Lettheequationoftheparabolabey2=4ax.LetPandQbethepoints(x1,y1)and(x2,y2).ThenthepolarofPisyy1=2a(x+x1).SincethispassesthroughQ(x2,y2),wegety1y2=2a(x1+x2).Thiscondition
showsthatthepoint(x1,y1)liesonthelineyy2=2a(x+x2).
∴ThepolarofQpassesthroughthepointP(x1,y1).
Example6.18
Pisavariablepointonthetangentatthevertexoftheparabolay2=4ax.ProvethatthelocusofthefootoftheperpendicularfromPonitspolarwithrespecttotheparabolaisthecirclex2+y2–2ax=0.
Solution
Pisthevariablepointonthetangentatthevertexoftheparabolay2=4ax.Theequationofthetangentatthevertexisx=0.AnypointonthetangentatthevertexisP(0,y1).Thepolarof(0,y)is
Theequationoftheperpendiculartothispolaris
Thispassesthrough(0,y1).
∴2ay1=k.∴TheequationoftheperpendicularfromPtoitspolaris
Let(l,m)bethepointofintersectionof(6.42)and(6.43).Then2al–my1=0.
y1l+2am–2ay1=0Solving
Now,
∴Thelocusof(l,m)isx2+y2–2ax=0.
Example6.19
Iffromthevertexoftheparabolay2=4ax,apairofchordscanbedrawnatrightanglestooneanotherandwiththesechordsasadjacentsidesarectanglebemade,provethatthelocusoffurtherangleoftherectangleistheparabolay2=4a(x–8a).
Solution
LetAPandAQbethechordsoftheparabolasuchthat .Completethe
rectangleAPRQ.ThenthemidpointsofARandPQarethesame.LettheequationsofAPbey=mx.Solvingy=mxandy2=4ax,wegetm2x2=4axor
∴ThepointPis .SinceAQisperpendiculartoAP,slopeofAQis .
Hence,thepointQis(4am2,−4am).
Let(x1,y1)bethepointR.ThemidpointofARis ThemidpointofPQis
SincethemidpointofARisthesameasthatofPQ,
Hence,thelocusof(x1,y1)isy2=4a(x–8a).
Example6.20
Showthatifr1andr2bethelengthsofperpendicularchordsofaparaboladrawn
throughthevertexthen
Solution
ThecoordinatesofPare(r1cosθ,r1sinθ).ThecoordinatesofQare(r2sinθ,r2cosθ).SincePliesontheparabolay2=4ax,
Similarly,
Also
Fromequations(6.45)and(6.46),
Example6.21
Showthatthelatusrectumofaparabolabisectstheanglebetweenthetangents
Showthatthelatusrectumofaparabolabisectstheanglebetweenthetangentsandnormalateitherextremity.
Solution
LetLSL′bethelatusrectumoftheparabolay2=4ax.ThecoordinatesofLare(a,2a).TheequationoftangentatLisy·2a=2a(x+a)
Theslopeofthetangentis1.∴TheslopeofthenormalatLis−1.LSisperpendiculartox-axis.
∴LatusrectumbisectstheanglebetweenthetangentsandnormalatL.
Example6.22
Showthatthelocusofthepointsofintersectionoftangentstoy2=4axwhichinterceptaconstantlengthdonthedirectrixis(y2–4ax)(x+a)2=d2x2.
Solution
LetP(x1,y1)bethepointofintersectiontangenttotheparabola.ThentheequationofthepairoftangentsPQandPRisT2=SS1.(i.e.)[yy1–2a(x+x1]2=(y2–4ax)(y2–4ax1).Whentheselinesmeetthe
directrixx=−a,wehave
Ify1andy2aretheordinatesofthepointofintersectionoftangentswiththedirectrixx+a=0,then
Then
∴Thelocusof(x1,y1)is(y2–4ax)(x+a)2=d2x2.
Example6.23
Showthatthelocusofmidpointsofchordsofaparabolawhichsubtendarightangleatthevertexisanotherparabolawhoselatusrectumishalfthelatusrectumoftheparabola.
Solution
Lettheequationoftheparabolabe
Let(x1,y1)bethemidpointofthechordPQ.ThentheequationofPQisT=S1
ThecombinedequationofthelinesAPandAQisgotbyhomogenizationofequation(6.48)withthehelpof(6.49).
∴ThecombinedequationofOPandOQis
Thelocusof(x1,y1)isy2=2a(x–4a)whichisaparabolawhoselatusrectumishalfthelatusrectumofthegivenparabola.
Example6.24
Showthatthelocusofmidpointsofchordsoftheparabolaofconstantlength2lis(y2–4ax)(y2+4a2)+4a2l2=0.
Solution
Let(x1,y1)bethemidpointofachordoftheparabola
Lettheequationofthechordbe
Anypointonthislineis(x1+rcosθ,y1+rsinθ).Thispointliesontheparabolay2=4ax.
ThetwovaluesofrarethedistancesRPandRQwhichareequalinmagnitudebutoppositeinsign.Theconditionforthisisthecoefficientofr=0.(i.e.)y1sinθ–2acosθ=0.
ThenfromEquation(6.50),
Thelocusof(x1,y1)is(y2–4ax)(y2+4a2)+4a2l2=0.(sincer=l)
Example6.25
Showthatthelocusofthemidpointsoffocalchordsofaparabolaisanotherparabolawhosevertexisatthefocusofthegivenparabola.
Solution
Letthegivenparabolabe
Let(x1,y1)bethemidpointofachordofthisparabola.Thenitsequationis
Ifthisisafocalchordthenthispassesthrough(a,0).
Thelocusof(x1,y1)isy2=2a(x–a)whichisaparabolawhosevertexisatthefocusofthegivenparabola.
Example6.26
Fromapointcommontangentsaredrawntothecircle andthe
parabolay2=4ax.Findtheareaofthequadrilateralformedbythecommontangents,thechordofcontactofthecircleandthechordofcontactoftheparabola.
Solution
Anytangenttotheparabolay2=4axis
Ifthisisalsoatangenttothecircle
∴m2(1+m2)=2orm4+m2–2=0or(m2–1)(m2+2)=0⇒m2=1or−2.Butm2=−2isinadmissiblesincem2hastobepositive.∴m
=±1.Hencethecommontangentsarey=±(x+a).ThetwotangentsmeetatP(−a,
0).
Theequationofthechordofcontactfrom(−a,0)tothecircle is
Theequationofthechordofcontactfrom(−a,0)totheparabolay2=4axis0=2a(x–a)orx–a=0.Whenx=a,y=±2a.HenceNandQare(a,2a)and(a,−2a).
When
∴AreaofquadrilateralLMQN=AreaoftrapeziumLMQN
Example6.27
ThepolarofapointPwithrespecttotheparabolay2=4axmeetsthecurveinQandR.ShowthatifPliesonthelinelx+my+n=0thenthelocusofthemiddlepointoftheQRisl(y2–4ax)+2a(lx+my+n)=0.
Solution
LetPbethepoint(h,k).ThepolarofP(h,k)withrespecttotheparabolay2=4axis
ThepolarofPmeetstheparabolay2=4axatQandR.LetP(x1,y1)bethemidpointofQR.Itsequationis
Equations(6.55)and(6.56)representthesameline.∴Identifyingequations(6.55)and(6.56)weget,
Sincethepoint(h,k)liesonlx+my+n=0,lh+mk+n=0.
Usingequation(6.56),
Example6.28
Provethatareaofthetriangleinscribedintheparabolay2=4axis
wherey1,y2andy3aretheordinatesoftheverticesofthe
triangle.
Solution
Let(x1,y1),(x2,y2)and(x3,y3)betheverticesofthetriangleinscribedinthe
parabolay2=4ax.Thentheverticesare .Theareaof
thetriangleis
Example6.29
Anequilateraltriangleisinscribedintheparabolay2=4axoneofwhoseverticesisatthevertexoftheparabola.Finditsside.
Solution
ThecoordinatesofBareB(rcos30°,rsin30°),
Sincethispointliesontheparabolay2=4ax,then
Exercises
1. Showthattwotangentscanbedrawnfromagivenpointtoaparabola.Ifthetangentsmakeanglesθ1andθ2withxaxissuchthat
i. tanθ1+tanθ2isaconstantshowthatthelocusofpointofintersectionoftangentsisastraightlinethroughthevertexofaparabola.
ii. iftanθ1·tanθ2isaconstantshowthatthelocusofthepointofintersectingisastraightline.
iii. ifθ1+θ2isaconstantshowthatthelocusofthepointofintersectionoftangentsisastraightlinethroughthefocus.
iv. ifθ1andθ2arecomplementaryanglesthenthelocusofpointofintersectionisthestraightlinex=a.
2. Findthelocusofpointofintersectionoftangentstotheparabolay2=4axwhichincludesanangle
of .
Ans.:3(x+a)2=y2–4ax
3. Showthatthelocusofthepolesofchordsoftheparabolay2=4axwhichsubtendsanangleof45°
atthevertexisthecurve(x+a)2=4(y2–4ax).
4. Showthatthelocusofpolesofalltangentstotheparabolay2=4axwithrespecttotheparabolay2
=4bxistheparabolaay2=4b2x.5. Showthatthelocusofpolesofchordsoftheparabolawhichsubtendsarightangleatthevertexisx+4a=0.
6. Showthatiftangentsbedrawntotheparabolay2=4axfromanypointonthestraightlinex+4a=0,thechordofcontactsubtendsarightangleatthevertexoftheparabola.
7. Perpendicularsaredrawnfrompointsonthetangentatthevertexontheirpolarswithrespectto
theparabolay2=4ax.Showthatthelocusofthefootoftheperpendicularisacirclecentreat(a,0)andradiusa.
8. Showthatthelocusofpoleswithrespecttotheparabolay2=4axoftangentstothecirclex2+y2
=4a2isx2–y2=4a2.9. ApointPmovessuchthatthelinethroughtheperpendiculartoitspolarwithrespecttothe
parabolay2=4axtouchestheparabolax2=4by.ShowthatthelocusofPis2ax+by+4a2x=0.
10. Ifachordoftheparabolay2=4axsubtendsarightangleatitsfocus,showthatthelocusofthe
poleofthischordwithrespecttothegivenparabolaisx2+y2+6ax+a2=0.
11. Showthatthelocusofpolesofallchordsoftheparabolay2=4axwhichareataconstantdistance
dfromthevertexisd2y2+4a2(d2–x2)=0.
12. Showthatthelocusofpolesofthefocalchordsoftheparabolay2=4axisx+a=0.
13. Iftwotangentstotheparabolay2=4axmakeequalangleswithafixedlineshowthatthechordofcontactpassesthroughafixedpoint.
14. Provethatthepolarofanypointonthecirclex2+y2–2ax–3a2=0withrespecttothecirclex2
+y2+2ax–3a2=0touchestheparabolay2=4ax.
15. Showthatthelocusofthepoleswithrespecttotheparabolay2=4axofthetangentstothecurve
x2–y2=aistheellipse4x2+y2=4ax.
16. Pisavariablepointontheliney=b,provethatthepolarofPwithrespecttotheparabolay2=4axisafixeddirectrix.
17. TheperpendicularfromapointOonitspolarwithrespecttoaparabolameetthepolarinthepointsMandcutstheaxisinG.Thepolarmeetsx-axisinTandtheordinatethroughOintersectsthecurveinPandP′.ShowthatthepointsG,M,P,P′andTlieonacirclewhosecentreisatthefocusS.
18. Tangentsaredrawntotheparabolay2=4axfromapoint(h,k).Showthattheareaofthetriangle
formedbythetangentsandthechordofcontactis
19. Provethatthelengthofthechordofcontactofthetangentsdrawnfromthepoint(x1,y1)tothe
parabolay2=4axis Henceshowthatoneofthetrianglesformedbythese
tangentsandtheirchordofcontactis
20. TangentsaredrawnfromavariablepointPtotheparabolay2=4axsuchthattheyformatriangle
ofconstantareawiththetangentatthevertex.ShowthatthelocusofPis(y2–4ax)x2=4c2.21. Provethatthetangenttoaparabolaandtheperpendiculartoitfromitsfocusmeetonthetangent
atthevertex.22. Showthataportionofatangenttoaparabolainterceptedbetweendirectrixandthecurvesubtends
arightangleatthefocus.
23. Thetangenttotheparabolay2=4axmakeanglesθ1andθ2withtheaxis.Showthatthelocusofthepointofintersectionsuchthatcotθ1+cotθ2=cisy=ac.
24. Ifperpendicularsbedrawnfromanytwofixedpointsontheaxisofaparabolaequidistantfromthefocusonanytangenttoit,showthatthedifferenceoftheirsquaresisaconstant.
25. Provethattheequationoftheparabolawhosevertexandfocusonx-axisatdistances4aand5a
fromtheoriginrespectively(a>0)isy2=4a(x–4a).Alsoobtaintheequationtothetangenttothiscurveattheendoflatusrectuminthefirstquadrant.
Ans.:y=x–a
26. Chordsofaparabolaaredrawnthroughafixedpoint.Showthatthelocusofthemiddlepointsisanotherparabola.
27. Findthelocusofthemiddlepointsofchordsoftheparabolay2=2xwhichtouchesthecirclex2+
y2–2x–4=0.
28. Atangenttotheparabolay2+4bx=0meetstheparabolay2=4axatPandQ.Showthatthelocus
ofthemiddlepointofPQisy2(2a+b)=4a2x.
29. Througheachpointofthestraightlinex–my=hisdrawnachordoftheparabolay2=4axwhich
isbisectedatthepoint.Provethatitalwaystouchestheparabola(y+2am)2=8axh.
30. Twolinesaredrawnatrightangles,onebeingatangenttotheparabolay2=4axandtheotherto
y2=4by.Showthatthelocusoftheirpointofintersectionisthecurve(ax+by)(x2+y2)=(bx–
ay)2.
31. Acirclecutstheparabolay2=4axatrightanglesandpassesthroughthefocus.Showthatthe
centreofthecircleliesonthecurvey2(a+x)=a(a+bx)2.32. Twotangentsdrawnfromapointtotheparabolamakeanglesθ1andθ2withthex-axis.Showthat
thelocusoftheirpointofintersectioniftan2θ1+tan2θ2=cisy
2–cx2=2ax.33. IfatrianglePQRisinscribedinaparabolasothatthefocusSistheorthocentreandthesidesmeet
theaxesinpointsK,LandMthenprovethatSK·SL·SM–4SA2=0whereAisthevertexoftheparabola.
34. Chordsoftheparabolay2=4axaredrawnthroughafixedpoint(h,k).Showthatthelocusofthe
midpointisaparabolawhosevertexis andlatusrectumis2a.
35. Showthatthelocusofthemiddlepointsofasystemofparallelchordsofaparabolaisalinewhichisparalleltotheaxisoftheparabola.
36. Showthatthelocusofthemidpointsofchordsoftheparabolawhichsubtendsaconstantangleα
atthevertexis(y2–2ax–8a2)2tan2α=16α2(4ax–y2).
ILLUSTRATIVEEXAMPLESBASEDONPARAMETERS
Example6.30
Provethatperpendiculartangentstotheparabolawillintersectonthedirectrix.
Solution
Letthetangentsatt1andt2intersectatP.Theequationoftangentsatt1andt2
are
Theslopesofthetangentsare .Sincethetangentsareperpendicular,
∴t1t2=−1
Thepointofintersectionofthetangentsatt1andt2isP(at1t2,a(t1+t2))(i.e.)
(−a,a(t1+t2)).Thispointliesonthelinex+a=0.∴Perpendiculartangentsintersectonthedirectrix.
Example6.31
Provethatthetangentsattheextremitiesofafocalchordintersectatrightanglesonthedirectrix.
Solution
Lett1andt2betheextremitiesofafocalchord.Thentheequationofthechordisy(t1+t2)=2x+2at1t2.Thispassesthroughthefocus(a,0).
∴Tangentsatt1andt2areperpendicular.Thepointofintersectionoftangentsatt1andt2is[at1t2,a(t1+t2)](i.e.)(−a,
a(t1+t2)).Thispointliesonthedirectrix.Hencethetangentsattheextremitiesofafocalchordintersectatrightanglesonthedirectrix.
Example6.32
Provethatanytangenttoaparabolaandperpendicularonitfromthefocusmeetonthetangentatthevertex.
Solution
Lettheequationoftheparabolay2=4ax.Theequationofthetangentattis
Theslopeofthetangentis .Theslopeoftheperpendiculartoitis–t.Hence
theequationoftheperpendicularlinepassingthroughfocus(a,0)is
Multiplyingequation(6.60)byt,weget
Equation(6.59)–equation(6.61)givesx(1+t2)=0orx=0.
∴y=at
Hence,thepointofintersectionof(6.59)and(6.60)is(0,at)andthispointliesony-axis.
Example6.33
Showthattheorthocentreofthetriangleformedbythetangentsatthreepointsonaparabolaliesonthedirectrix.
Solution
Lett1,t2andt3bepointsofcontactofthetangentsatthepointsA,BandC,respectivelyontheparabolay2=4ax,formingatrianglePQR.TheequationofQRis
Pisthepointofintersectionoftangentsatt1andt2.Thispointis[at1t2,a(t1+t2)].TheslopeofPL,perpendiculartoQRis−t3.∴TheequationofPLisy–a(t1+t2)=–t3[x–at1t2]
(i.e.)
ThentheequationofQMperpendicularfromQonPRis
Equation(6.63)–equation(3)givesx(t3–t2)=a(t2–t3)orx=−a.Thispointliesonthedirectrixx+a=0.Hencetheorthocentreliesonthe
directrix.
Example6.34
Thecoordinatesoftheendsofafocalchordoftheparabolay2=4axare(x1,y1)and(x2,y2).Provethatx1x2=a2andy1y2=−4a2.
Solution
Lett1andt2betheendsofafocalchord.Thentheequationofthefocalchordisy(t1+t2)=2x+at1t2.Sincethispassesthroughthefocus(a,0),0=2a+at1t2ort1t2=−1.
Example6.35
Aquadrilateralisinscribedinaparabolaandthreeofitssidespassthroughfixedpointsontheaxis.Showthatthefourthsidealsopassesthroughafixedpointontheaxisoftheparabola.
Solution
Lett1,t2,t3andt4berespectivelyverticesA,B,CandDofthequadrilateralinscribedintheparabolay2=4ax.TheequationofchordABis
Whenthismeetsthex-axisy=0(i.e.)x=−at1t2=k1.SinceABmeetsthex-axisatafixedpoint,
Similarly,
Multiplyingthese,weget
Hence,thefourthsideofthequadrilateralalsopassesthroughafixedpoint.
Example6.36
Tangentstotheparabolay2=4axaredrawnatpointswhoseabscissaeareintheratiok:1.Provethatthelocusoftheirpointofintersectionisthecurvey2=(k1/4
+k–1/4)2x2.
Solution
Letthetangentsatt1andt2intersectatP(x1,y1)
Giventhat
Thepointofintersectionofthetangentsatt1andt2isx1=at1t2andy1=a(t1+t2).
∴Thelocusof(x1,y1)isy2=(k1/4+k−1/4)2x.
Example6.37
Showthatthelocusofthemiddlepointofalltangentsfrompointsonthedirectrixtotheparabolay2=4axisy2(2x+a)=a(x+3a)2.
Solution
Let(−a,y1)beapointonthedirectrix.Lettbethepointofcontactoftangentsfrom(−a,y1)totheparabolay2=4ax.Theequationofthetangentattis
Sincethispassesthrough(−a,y1),
∴Thepointonthedirectrixis
Let(x1,y1)bethemidpointoftheportionoftangentbetweenthedirectrixandthepointofcontact.Then
and2y1t=2at2–a+at2
Thelocusof(x1,y1)isy2(2x+a)=a(3x+a)2.
Example6.38
TangentsaredrawnfromavariablepointPtotheparabolay2=4ax,suchthattheyformatriangleofconstantareac2withthetangentatthevertex.ShowthatthelocusofPis(y2–4ax)x2=4c4.
Solution
LetP(x1,y1)bethepointofintersectionoftangentsatt1andt2.Theequationof
thetangentatt1is ThismeetsthetangentatthevertexatQ.∴Qis
(0,at1).Similarly,Ris(0,at2).Pisthepointofintersectionoftangentsatt1andt2andthepointisP(at1t2,a(t1+t2)).TheareaofΔPQRisgivenasc2.
Therefore,thelocusof(x1,y1)isx2(y2–4ax)=4c4.
Example6.39
Provethatthedistanceofthefocusfromtheintersectionoftwotangentstoaparabolaisameanproportionaltothefocalradiiofthepointofconstant.
Solution
Letthetangentsat intersectatP.Thenthe
coordinatesofthepointPare(at1t2,a(t1+t2)).Sisthepoint(a,0).
Example6.40
Provethatthelocusofthepointofintersectionofnormalsattheendsofafocalchordofaparabolaisanotherparabolawhoselatusrectumisonefourthofthatofthegivenparabola.
Solution
Lettheequationoftheparabolabe
Lett1andt2betheendsofafocalchordoftheparabola.Forafocalchordt1t2=−1.
Theequationofthenormalatt1andt2are
If(x1,y1)isapointofintersectionofthenormalsatt1andt2then
Thelocusof(x1,y1)isy2=a(x–3a)whichisaparabolawhoselatusrectumisonefourthofthelatusrectumoftheoriginalparabola.
Example6.41
Ifthenormalatthepointt1ontheparabolay2=4axmeetsthecurveagainatt2
provethat
Solution
Theequationofthenormalatt1is
Theequationofthechordjoiningthepointst1andt2is
Equations(6.73)and(6.74)representthesamelines.Thereforeidentifyingwe
get
Example6.42
Ifthenormalsattwopointst1,t2ontheparabolay2=4axintersectagainatapointonthecurveshowthatt1+t2+t3=0andt1t2=2andtheproductofordinatesofthetwopointsis8a2.
Solution
Thenormalst1andt2meetatt3.
Subtracting Sincet1–t2≠0,t1t2=2.
Solvingequations(6.75)and(6.76),weget
Example6.43
Findtheconditionthatthelinelx+my+n=0isanormaltotheparabolaisy2=4ax.
Solution
Letthelinelx+my+n=0beanormalat‘t’.Theparabolaisy2=4ax.Theequationofthenormalattis
Buttheequationofthenormalisgivenas
Identifyingequations(6.77)and(6.78),weget and
(i.e.)al3+2alm2+m2n=0.
Example6.44
Showthatthelocusofpolesofnormalchordsoftheparabolaisy2=4axis(x+2a)y2+4a3=0.
Solution
Let(x1,y1)bethepoleofanormalchordnormalatt.Theequationofthepolarof(x1,y1)is
Theequationofthenormalattis
Equations(6.79)and(6.80representthesameline.∴Identifyingequations(6.79)and(6.80),weget
Example6.45
Intheparabolay2=4axthetangentatthepointPwhoseabscissaisequaltothelatusrectummeetstheaxisonTandthenormalatPcutsthecurveagaininQ.ProvethatPT:TQ=4:5.
Solution
LetPandQbethepointst1andt2respectively.Giventhat
Theequationofthetangentatt1is
whenthismeetsthex-axis,y=0.
HenceTisthepoint(−4a,0).Alsoasthenormalatt1meetsthecurveatt2,
Example6.46
Showthatthelocusofapointsuchthattwoofthethreenormaldrawnfromittotheparabolay2=4axcoincideis27ay2=4(x–2a)3.
Solution
Let(x1,y1)beagivenpointandtbefootofthenormalfrom(x1,y1)totheparabolay2=4ax.Theequationofthenormalat‘t’is
Sincethispassesthrough(x1,y1)wehave
Ift1,t2andt3befeetofthenormalsfrom(x1,y1)totheparabolathent1,t2andt3aretherootsofequation(6.84).
Iftwoofthethreenormalscoincidethent1=t2.
Fromequations(6.88)and(6.89), Sincet1isarootof
equation(6.84)
Example6.47
Ifthenormalsfromapointtotheparabolay2=4axcuttheaxisinpointswhosedistancesfromthevertexareinAPthenshowthatthepointliesonthecurve
27ay2=2(x–2a)3.
Solution
Let(x1,y1)beagivenpointandtbethefootofanormalfrom(x1,y1).Theequationofthenormalattis
Sincethispassesthrough(x1,y1),y1+x1t=2at+at3.
Ift1,t2andt3bethefeetofthenormalsfrom(x1,y1)then
Whenthenormalattmeetsthex-axis,y=0,from(6.91)wegetxt=2at+at3
orx=2a+at2.Thenthex-coordinatesofthepointswherethenormalmeetsthex-axisare
givenby GiventheseareinAP.
areinAP.
Fromequation(6.95),or
Sincet2isarootofequation(6.91),
∴Thelocusof(x1,y1)is27ay3=2(x–2a)3.
Example6.48
Showthatthelocusofthepointofintersectionoftwonormalstotheparabolawhichareatrightanglesisy2=a(x–3a).
Solution
If(x1,y1)isthepointofintersectionoftwonormalstotheparabolay2=4axthen
Ift1,t2andt3bethefeetofthethreenormalsfrom(x1,y1)then
Sincetwoofthenormalsareperpendicularthent1t2=−1
Sincet3isarootofequation(6.97),
Example6.49
Provethatanormalchordofaparabolawhichsubtendsarightangleatthe
vertexmakesanangle withthex-axis.
Solution
Lettheequationoftheparabolabe
Theequationofthenormalattis
ThecombinedequationofthelinesAPandAQis
Sincethetwolinesareatrightangles,coefficientofx2+coefficientofy2=0.
∴t=0ort2=2.t=0correspondstothenormaljoiningthroughthevertex.
∴Thenormalsmakeanangle withthex-axis.
Example6.50
Provethattheareaofthetriangleformedbythenormalstotheparabolay2=4ax
atthepointst1,t2andt3is
Solution
Theequationsofthenormalsatt1,t2,t3are
Solvingtheseequationspairwisewegettheverticesofthetriangle.Hencethe
verticesare andtwoothersimilarpoints.
Example6.51
ProvethatthelengthoftheinterceptsonthenormalatthepointP(at2,2at)totheparabolay2=4axmadebythecircledescribedonthelinejoiningthefocusand
Pasdiameteris
Solution
TheequationofthenormalatPisy+xt=2at+at3.
LetthecircleonPSasdiametercutthenormalatPatRandthex-axisatT.
Example6.52
NormalsatthreepointsP,QandRoftheparabolay2=4axmeetin(h,k).Prove
thatthecentroidofΔPQRliesontheaxisatadistance fromthevertex.
Solution
Lettbeafootofanormalfrom(h,k).Theequationofthenormalattis
Thispassesthrough(h,k).
⇒at3+t(2a–h)–k=0.Ift1,t2andt3arethefeetofthenormalsfrom(h,k)thent1+t2+t3=0,
ThecentroidoftheΔPQRis
Sincethecentroidliesonthex-axis,
Thex-coordinatesofthecentroidis
∴Centroidisatadistance fromthevertexoftheparabola.
Example6.53
ThenormalsatthreepointsP,QandRonaparabolameetatTandSbethefocusoftheparabola.ProvethatSP·SQ·SR=aTS2.
Solution
LetTbethepoint(h,k).ThenP,QandRarethefeetofthenormalsfromT(h,k).Theequationofthenormalattisy+xt=2at+at3.Ift1,t2andt3bethefeetofthenormalsfromTthent1+t2+t3=0.
Sisthepoint(a,0).
Example6.54
TheequationofachordPQoftheparabolay2=4axislx+my=1.Showthat
thenormalsatP,Qmeetonthenormalat
Solution
LetPandQbethepointst1andt2.ThenormalsatPandQmeetatR.Ift3isthefootofthenormalofthe3rdpointthen
TheequationofthechordPQislx+my=1.SincePandQarethepointst1andt2,
Example6.55
IfthenormalatPtotheparabolay2=4axmeetsthecurveatQandmakeanangleθwiththeaxisshowthat
i. itwillcuttheparabolaatθatanangle and
ii. PQ=4asecθcosec2θ.
Solution
LetPbethepoint(at2,2at).Theequationofthenormalattisy+xt=2at+at3.
Thenormalattmeetsthecurveat
LetɸbetheanglebetweenthenormalandthetangentatQ.Theslopeofthe
tangentatQis
Slopeofthenormalattis–t.
sincetanθ=−t
Example6.56
Provethatthecirclepassingthroughthefeetofthethreenormalstoaparaboladrawnfromanypointintheplanepassesthroughthevertexoftheparabola.Alsofindtheequationofthecirclepassingthroughthefeetofthenormals.
Solution
Lettheequationoftheparabolabe
Lettheequationofthecirclebe
LetP,QandRbethefeetofthenormalstoy2=4axfromagivenpoint(h,k).Thenwehaveat3+(2a–h)t–k=0.
Ift1,t2andt3bethefeetofthenormalsatP,QandRthent1+t2+t3=0.Weknowthatthecircle(6.111)andtheparabola(6.110)cutatfourpointsandift1,t2,t3andt4arethefourpointsofintersectionofthecircleandtheparabolathentheyaretherootsoftheequation,
Ift1,t2andt3correspondtothefeetofthenormalsfrom(h,k)then
Fromequations(6.113)and(6.114),t4=1.
Butt4isthepoint whichisthevertexoftheparabola.Hence
thecirclepassingthroughthefeetofthenormalsfromagivenpointalsopassesthroughthevertexoftheparabola.Henceequation(6.112)becomes
sincec=0Equations(6.111)and(6.115)arethesame.Bycomparingthecoefficients,
weget
∴Theequationofthecirclepassingthroughthefeetofthenormalis
Exercises
1. Showthattheportionofthetangentinterceptedbetweenthepointofcontactandthedirectrix
subtendsarightangleatthefocus.2. IfthetangentatapointPontheparabolameetstheaxisatTandPNistheordinateatPthenshow
thatAN=AT.3. IfthetangentatPmeetsthetangentatthevertexinYthenshowthatSYisperpendiculartoTPand
SY2=ASSP.4. IfA,BandC,arethreepointsonaparabolawhoseordinatesareinGPthenprovethatthetangents
atAandCmeetontheordinatesofB.5. Provethatthemiddlepointoftheinterceptsmadeonatangenttoaparabolabythetangentsattwo
pointsPandQliesonthetangentwhichisparalleltoPQ.
6. Ifpoints(at2,2at)isoneextremityofafocalchordoftheparabolay2=4ax,showthatthelength
ofthefocalchordis
7. Showthatthetangentsatoneextremityofafocalchordofaparabolaisparalleltothenormalattheotherextremity.
8. Ifthetangentsatthreepointsontheparabolay2=4axmakeangles60°,45°and30°withtheaxisoftheparabola,showthattheabscissaeandordinatesofthethreepointsareinGP.
9. Showthatthecircledescribedonthefocalchordofaparabolaasdiametertouchesthedirectrix.10. Showthatthetangentatoneextremityofafocalchordofaparabolaisparalleltothenormalatthe
otherextremity.11. Provethatthesemilatusrectumofaparabolaistheharmonicmeanofthesegmentsofafocal
chord.12. Provethatthecircledescribedonfocalradiiasdiametertouchesthetangentsatthevertexofa
parabola.
13. Threenormalstoaparabolay2=4xaredrawnthroughthepoint(15,12).Showthattheequationsare3x–y–33=0,4x+y–72=0andx–y–3=0.
14. ThenormalsattwopointsPandQofaparabolay2=4axmeetatthepoint(x1,y1)ontheparabola.ShowthatPQ=(x1+4a)(x1–8a).
15. Showthatthecoordinatesofthefeetofthenormalsoftheparabolay2=4axdrawnfromthepoint(6a,0)are(0,0),(4a,4a)and(4a,–4a).
16. ThenormalatPtotheparabolay2=4axmakesanangleawiththeaxis.Showthattheareaofthetriangle,formedbyitisthetangentsatitsextremitiesisaconstant.
17. IfP,QandRarethepointst1,t2andt3ontheparabolay2=4ax,suchthatthenormalatQandR
meetatPthenshowthat:i. thelinePQispassesthroughafixedpointontheaxis.ii. thelocusofthepoleofPQisx=a.
iii. thelocusofthemidpointofPQisy2=2a(x+2a).
iv. theordinatesofPandQaretherootsoftheequationy2+xy+8a2=0wheret3istheordinateofthepointofintersectionofthenormalsatPandQ.
18. IfacirclecutsaparabolaatP,Q,RandSshowthatPQandRSareequallyinclinedtotheaxis.
19. ThenormalsatthepointsPandRontheparabolay2=4axmeetontheparabolaatthepointP.
ShowthatthelocusoftheorthocentreofΔPQRisy2=a(x+6a)andthelocusofthecircumcentre
ofΔPQRistheparabola2y=x(x–a).20. Provethattheareaofthetriangleinscribedinaparabolaistwicetheareaofthetriangleformedby
thetangentsatthevertices.21. ProvethatanythreetangentstoaparabolawhoseslopesareinHPenclosesatriangleofconstant
area.22. Provethatthecircumcircleofatrianglecircumscribingaparabolapassesthroughthefocus.
23. IfthenormalsatanypointPoftheparabolay2=4axmeettheaxisatGandthetangentatvertexatHandifAbethevertexoftheparabolaandtherectangleAGQHbecompleted,provethatthe
equationtothelocusofQisx2=2ax+ay2.24. ThenormalatapointPofaparabolameetsthecurveagainatQandTisthepoleofPQ.Show
thatTliesonthedirectrixpassingthroughPandthatPTisbisectedbythedirectrix.
25. Iffromthevertexoftheparabolay2=4ax,apairofchordsbedrawnatrightanglestooneanotherandwiththesechordsasadjacentsidesarectanglebemadethenshowthatthelocusoffurther
angleoftherectangleistheparabolay2=4a(x–8a).
26. Thenormaltotheparabolay2=4axatapointPonitmeetstheaxisinG.ShowthatPandGareequidistantfromthefocusoftheparabola.
27. Twoperpendicularstraightlinesthroughthefocusoftheparabolay2=4axmeetitsdirectrixinTandT′respectively.ShowthatthetangentstotheparabolatotheperpendicularlinesintersectatthemidpointofTT′.
28. IfthenormalsatanypointP(18,12)totheparabolay2=8xcutsthecurveagainatQshowthat9·
PQ=80
29. IfthenormalatPtotheparabolay2=4axmeetsthecurveagainatQandifPQandthenormalat
Qmakeanglesθandɸ,respectivelywiththeaxis,provethattanθtan2ɸ+tan2θ+2=0.30. PQisafocalchordofaparabola.PP′andQQ′arethenormalsatPandQcuttingthecurveagain
atP′andQ′.ShowthatP′Q′isparalleltoPQandisthreetimesPQ.
31. IfPQbeanormalchordoftheparabola.y2=4axandifSbethefocus,showthatthelocusofthe
centroidofthetriangleSPQisy2(ay2+180a2–108ax)+128a4=0.32. IfthetangentsatPandQmeetatTandtheorthocenteroftheΔPTQliesontheparabola,show
thateithertheorthocentreisatthevertexorthechordPQisnormaltotheparabola.
33. Ifthreenormalsfromapointtotheparabolay2=4axcutstheaxisinpoints,whosedistancesfrom
thevertexareinAP,showthatthepointonthecurve27ay2=2(x–a)3.34. Tangentsaredrawntoaparabolafromanypointonthedirectrix.Showthatthenormalsatthe
pointsofcontactareperpendiculartoeachotherandthattheyintersectonanotherparabola.
35. Showthatiftwotangentstoaparabolay2=4axinterceptaconstantlengthonanyfixedtangent,thelocusoftheirpointofintersectionisanotherequalparabola.
36. Showthattheequationofthecircledescribedonthechordinterceptedbytheparabolay2=4axon
theliney=mx+casdiameterism2(x2+y2)+2(mc–2a)x–4ay+c(4am+c)=0.37. Circlesaredescribedonanytwocommonchordsofaparabolaasdiameter.Provethattheir
commonchordpassesthroughthevertexoftheparabola.
38. IfP(h,k)isafixedpointintheplaneofaparabolay2=4ax.ThroughPavariablesecantisdrawn
tocuttheparabolainQandR.TisapointonQRsuchthat
i. PQ·PR=PT2.ShowthatthelocusofTis(y–k)2=k2–4ah.
ii. PQ+PR=PT.ShowthatthelocusofTisy2–k2=4a(x–h).
39. Showthatthelocusofthepointofintersectionoftangents,totheparabolay2=4axatpoints
whoseordinatesareintheratio
40. Showthatthelocusofthemiddlepointsofasystemofparallelchordsofaparabolaisalinewhichisparalleltotheaxisoftheparabola.
41. P,QandRarethreepointsonaparabolaandthechordPQmeetsthediameterthroughRinT.
OrdinatesPMandQNaredrawntothisdiameter.ShowthatRMRN=RT2.
Chapter7
Ellipse
7.1STANDARDEQUATION
Aconicisdefinedasthelocusofapointsuchthatitsdistancefromafixedpointbearsaconstantratiotoitsdistancefromafixedline.Thefixedpointiscalledthefocusandthefixedstraightlineiscalledthedirectrix.Theconstantratioiscalledtheeccentricityoftheconic.Iftheeccentricityislessthanunitytheconiciscalledanellipse.Letusnowderivethestandardequationofanellipseusingtheabovepropertycalledfocus-directrixproperty.
7.2STANDARDEQUATIONOFANELLIPSE
LetSbethefocusandlinelbethedirectrix.DrawSXperpendiculartothedirectrix.DivideSXinternallyandexternallyintheratioe:1(e<1).LetAandA′
bethepointsofdivision.Since and ,fromthedefinitionofellipse,
thepointsAandA′lieontheellipse.LetAA′=2aandCbeitsmiddlepoint.
Addingequations(7.1)and(7.2),wegetSA+SA′=e(AX+A′X).
Subtractingequations(7.1)from(7.2),wegetSA′−SA=e(CX′−CX)
TakeCSasthex-axisandCMperpendiculartoCS,asy-axis.LetP(x,y)beanypointontheellipse.DrawPMperpendiculartothe
directrix.ThenthecoordinatesofSare(ae,0).Fromthefocus-directrixproperty
oftheellipse,
Thisiscalledthestandardequationofanellipse.
Note7.2.1:
1. Equation(7.5)canbewrittenas:
2. AA′iscalledthemajoraxisoftheellipse.3. BB′iscalledtheminoraxisoftheellipse.4. Ciscalledthecentreoftheellipse.5. Thecurvemeetsthex-axisatthepointA(a,0)andA′(−a,0).6. Thecurvemeetsthey-axisatthepointsB(0,b)andB′(0,−b).7. Thecurveissymmetricalaboutboththeaxes.If(x,y)isapointonthecurve,then(x,−y)and(−x,y)arealsothepointsonthecurve.
8. Fromtheequationoftheellipse,weget
Therefore,foranypoint(x,y)onthecurve,−a≤x≤aand−b≤y≤b.9. Thedoubleordinatethroughthefocusiscalledthelatusrectumoftheellipse.
(i.e.)LSL′isthelatusrectum.
10. Secondfocusandseconddirectrix:Onthenegativesideoftheorigin,takeapointS′suchthatCS=CS′andanotherpointX′suchthatCX=CX′=a.
DrawX′M′perpendiculartoAA′andPM′perpendiculartoX′M′.Thenwecanshowthat
givesthelocusofPas HereS′iscalledthesecondfocusandX′M′isthesecond
directrix.11.
i. ShiftingtheorigintothefocusS,theequationoftheellipseis
ii. ShiftingtheorigintoA,theequationoftheellipseis
iii. ShiftingtheorigintoX,theequationofthefocusis
12. Theequationofanellipseiseasilydeterminedifwearegiventhefocusandtheequationofthedirectrix.
7.3FOCALDISTANCE
Thesumofthefocaldistancesofanypointontheellipseisequaltothelengthofthemajoraxis.
Intheabovefigure,(section2.2)
Note7.3.1:
7.4POSITIONOFAPOINT
Apoint(x1,y1)liesinside,onoroutsideoftheellipseaccordingas −1
isnegative,zeroorpositive.
LetQ(x1,y1)beapointontheordinatePNwherePisapointontheellipse
Then,
Similarly,iftheQ′(x′,y′)isapointoutsidetheellipse,
EvidentlyifQ(x′,y′)isapointontheellipse,
7.5AUXILIARYCIRCLE
Thecircledescribedonthemajoraxisasdiameteriscalledtheauxiliarycircle.LetPbeanypointontheellipse.LettheordinatethroughPmeettheauxiliary
circleatP′.Since wehavethegeometricalrelation,P′N2=AN·A′N.
ThepointP′wheretheordinatePNmeetstheauxiliarycircleiscalledthecorrespondingpointofP.Therefore,theordinateofanypointontheellipsetothatofcorrespondingpointontheellipseareintheratiosoflengthsofsemi-minoraxisandsemi-majoraxis.Thisratiogivesanotherdefinitiontoanellipse.Consideracircleandfromeachpointonit,drawperpendiculartoadiameter.Thelocusofthesepointsdividingtheseperpendicularsinagivenratioisan
ellipseandforthisellipsethegivencircleistheauxiliarycircle.
ILLUSTRATIVEEXAMPLESBASEDONFOCUS-DIRECTRIXPROPERTY
Example7.1
Findtheequationoftheellipsewhosefoci,directrixandeccentricityaregivenbelow:
i. Focusis(1,2),directrixis2x−3y+6=0andeccentricityis2/3ii. Focusis(0,0),directrixis3x+4y−1=0andeccentricityis5/6
iii. Focusis(1,–2),directrixis3x−2y+1=0andeccentricityis1/
Solution
i. LetP(x1,y1)beapointontheellipse.Then
Therefore,thelocusof(x1,y1)istheellipse101x2+81y2+48x−330x−324y+441=0.
ii.
Therefore,thelocusof(x1,y1)istheellipse27x2+20y2−24xy+6x+8y−1=0.
iii.
Therefore,thelocusof(x1,y1)istheellipse17x2+22y2+12xy−58x+108y+129=0.
Example7.2
Findtheequationoftheellipsewhose
i. Fociare(4,0)and(−4,0)and
ii. Fociare(3,0)and(−3,0)and
Solution
i. Ifthefociare(ae,0)and(−ae,0)thentheequationoftheellipseis Here,ae=4and
∴Theequationoftheellipseis
ii. Ifthefociare(ae,0)and(−ae,0)theequationoftheellipseis
Here,ae=3and a2e2=9and
Therefore,theequationoftheellipseis
Example7.3
Findtheeccentricity,fociandthelengthofthelatusrectumoftheellipse.
i. 9x2+4y2=36
ii. 3x2+4y2−12x−8y+4=0
iii. 25x2+9y2−150x−90y+225=0.
Solution
i. 9x2+4y2=36Dividingby36,weget
Thisisanellipsewhosemajoraxisisthey-axisandminoraxisisthex-axisandcentreattheorigin.
Therefore,eccentricity=
Therefore,fociare
Therefore,latusrectum=
ii.
Shifttheorigintothepoint(2,1).Therefore,centreis(2,1).
Therefore,theequationoftheellipseis
Therefore,fociare(3,1)and(1,1)withrespecttooldaxes.
Lengthofthelatusrectum
iii.
Shifttheorigintothepoint(3,5).
Therefore,theequationoftheellipseis
Therefore,centreis(3,5).Thisisanellipsewithy-axisonthemajoraxisandx-axisastheminoraxis.
Therefore,focilieonthelinex=3.
Therefore,fociare(3,9)and(3,1)andLeangthofthelatusrectum=
Exercises
1. Findthecentre,fociandlatusrectumoftheellipse:
i. 3x2+4y2+12x+8y−32=0Ans.:(−2,−1);(0,−1);(−4,−1);6
ii. 9x2+25y2=225
Ans.:
iii. x2+9y2=9
Ans.:
iv. 2x2+3y2−4x+6y+4=0
Ans.:
2. Findtheequationoftheellipsewhosefociare(0,±2)andthelengthofmajoraxisis2
Ans.:5x2+y2=5
3. Findtheequationoftheellipsewhosefociis(3,1),eccentricity anddirectrixisx−y+6=0.
Ans.:7x2+2xy+7y2−60x−20y+44=0
4. Findtheequationofellipsewhosecentreisattheorigin,onefocusis(0,3)andthelengthofsemi-majoraxisis5.
Ans.:
5. Findtheequationofellipsewhosefocusis(1,−1),eccentricityis anddirectrixisx−y+3=0.
Ans.:7x2+2xy+7y2−22x+22y+7=0
6. Findtheequationoftheellipsewhosecentreis(2,−3),onefocusat(3,−3)andonevertexat(4,−3).
Ans.:3x2+4y2−12x+24y+36=0
7. Findthecoordinatesofthecentre,eccentricityandfocioftheellipse8x2+6y2−6x+12y+13=0
Ans.:
8. Findtheequationoftheellipsewithfociat(0,1)and(0,−1)andminoraxisoflength1.
Ans.:2x2+4y2=5
9. Anellipseisdescribedbyusingoneendlessstringwhichispassedthroughtwopoints.Iftheaxesare6and4unitsfindthenecessarylengthandthedistancebetweenthepoints.
Ans.:
7.6CONDITIONFORTANGENCY
Tofindtheconditionthatthestraightliney=mx+cmaybeatangenttotheellipse:
Lettheequationoftheellipsebe
Lettheequationofthestraightlinebe
Solvingequations(7.6)and(7.7),wegettheirpointsofintersection;thex-
coordinatesofthepointsofintersectionaregivenby
Ify=mx+cisatangenttotheellipsethenthetwovaluesofxofthisequationareequal.Theconditionforthatisthediscriminantofthequadraticequationiszero.
Thisistherequiredconditionfortheliney=mx+ctobeatangenttothegivenellipse.
Note7.6.1:Theequationofanytangenttotheellipseisgivenby
7.7DIRECTORCIRCLEOFANELLIPSE
Toshowthatalwaystwotangentscanbedrawnfromagivenpointtoanellipseandthelocusofpointofintersectionofperpendiculartangentsisacircle:
Lettheequationoftheellipsebe
Anytangenttothisellipseis
Ifthistangentpassesthroughthepoint(x1,y1)theny1=
Thisisaquadraticequationinmandhencetherearetwovaluesform.Foreachvalueofm,thereisatangent(realorimaginary)andhencetherearetwo
tangentsfromagivenpointtoanellipse.Ifm1andm2aretherootsofthe
equation(7.11),then
Ifthetwotangentsareperpendicularthenm1m2=−1.
Thelocusof(x1,y1)isx2+y2=a2+b2whichisacircle,centreat(0,0)and
radius
Note7.7.1:Thiscircleiscalledthedirectorcircleoftheellipse.
7.8EQUATIONOFTHETANGENT
Tofindtheequationofthechordjoiningthepoints(x1,y1)and(x2,y2)andfindtheequationofthetangentat(x1,y1)totheellipse:LetP(x1,y1)andQ(x2,y2)betwopointsontheellipse.Lettheequationof
ellipsebe
Then
and
Subtracting,
Fromequation(7.15),wegettheequationofthechordjoiningthepoints(x1,y1)and(x2,y2)as:
Thischordbecomesthetangentat(x1,y1)ifQtendstoPandcoincideswithP.Hence,byputtingx2=x1andy2=y1inequation(7.16),wegettheequationofthetangentat(x1,y1).Therefore,theequationofthetangentat(x1,y1)is:
Dividingbya2b2,weget
However, since(x1,y1)liesontheellipse.
Therefore,fromequation(7.17),theequationofthetangentat(x1,y1)is
7.9EQUATIONOFTANGENTANDNORMAL
Tofindtheequationoftangentandnormalat(x1,y1)totheellipse
Theequationoftheellipseis
Differentiatingwithrespecttox,weget
However, =slopeofthetangentat(x1,y1).Therefore,theequationof
thetangentat(x1,y1)is,
Dividingbya2b2,weget
Slopeofthenormalat(x1,y1)is
Theequationofthenormalat(x1,y1)totheellipseis
Dividingbyx1,y1,weget,
Therefore,theequationofnormalat(x1,y1)totheellipse is
7.10EQUATIONTOTHECHORDOFCONTACT
Tofindtheequationtothechordofcontactoftangentsdrawnfrom(x1,y1)
totheellipse
Theequationoftheellipseis
LetQRbethechordofcontactoftangentsfromP(x1,y1).LetQandRbethepoints(x2,y2)and(x3,y3),respectively.ThentheequationoftangentsatQandR
are:
ThesetwotangentspassthroughP(x1,y1).
Therefore, and
Theabovetwoequationsshowthatthepoints(x2,y2)and(x3,y3)lieontheline
Hence,theequationofthechordofcontactis
7.11EQUATIONOFTHEPOLAR
TofindtheequationofthepolarofthepointP(x1,y1)ontheellipse
LetP(x1,y1)bethegivenpoint.LetQRbeavariablechordthroughthepointP(x1,y1).LetthetangentsatQandRmeetatT(h,k).TheequationofthechordcontactfromT(h,k)is:
Thischordofcontactpassesthrough(x1,y1).
ThelocusofT(h,k)isthepolarofthepoint(x1,y1).
Therefore,thepolarof(x1,y1)is
Note7.11.1:
1. Whenthepoint(x1,y1)liesontheellipse,thepolarof(x1,y1)isthetangentat(x1,y1).Whenthepoint(x1,y1)liesinsidetheellipsethepolarof(x1,y1)isthechordofcontactoftangentsfrom(x1,y1).
2. Theline iscalledthepolarofthepoint(x1,y1)and(x1,y1)iscalledthepoleofthe
line
7.12CONDITIONFORCONJUGATELINES
Tofindthepoleofthelinelx+my+n=0withrespecttotheellipse
anddeducetheconditionforthelineslx+my+n=0andl1x+m1y
+n1=0tobeconjugatelines:
Let(x1,y1)bethepoleoftheline
withrespecttotheellipse
Thenthepolarof(x1,y1)is:
Thentheequations(7.24)and(7.26)representthesameline.∴Identifyingequations(7.24)and(7.26),weget
Hence,thepoleofthelinelx+my+n=0is Twolinesaresaidto
beconjugateifthepoleoftheeachliesontheother.
∴Thepoint liesonthelinel1x+m1y+n1=0.
Thisistherequiredconditionforthelineslx+my+n=0andl1x+m1y+n1=0tobeconjugatelines.
ILLUSTRATIVEEXAMPLESBASEDONTANGENTS,NORMALS,POLE-POLARANDCHORD
Example7.4
Findtheequationofthetangenttotheellipsex2+2y2=6at(2,−1).
Solution
Theequationoftheellipseisx2+2y2=6.
Theequationofthetangentat(x1,y1)is
Therefore,theequationofthetangentat(2,−1)is
(i.e.)2x−2y=6⇒x−y=3
Example7.5
Findtheequationofthenormaltotheellipse3x2+2y2=5at(−1,1).
Solution
Therefore,theequationofthenormaltotheellipse3x2+2y2=5is2x+3y=1.
Example7.6
IfBandB′aretheendsoftheminoraxisofanellipsethenprovethatSB=S′B′=awhereSandS′arethefocianda′isthesemi-majoraxis.Showalsothat
SBS′B′isarhombuswhoseareais2abe.
Solution
Sis(ae,0);S′is(−ae,0)Bis(0,b);B′is(0,−b)
Inthefigure,SBS′B′thediagonalsSS′andBB′areatrightangles.Therefore,SBS′B′isarhombus.
Example7.7
IfthetangentatPoftheellipse meetsthemajoraxisatTandPNisthe
ordinateofP,thenprovethatCN·CT=a2whereCisthecentreoftheellipse.
Solution
LetPbethepoint(x1,y1)
Theequationoftangentat(x1,y1)is
Whenthetangentmeetsthex-axis,y=0
Example7.8
ThetangentatanypointPontheellipse meetsthetangentsatAandA′
(extremitiesofmajoraxis)inLandM,respectively.ProvethatAL·A′M=b2.
Solution
LettheequationofthetangentatPbe Theequationofthe
tangentatthepointAisx=a.Solvingthesetwoequations,weget
Example7.9
IfSYandS′Y′beperpendicularsfromthefociuponthetangentsatanypointof
theellipse ,thenprovethatY,Y′lieonthecirclex2+y2=a2andthatSY
·S′Y′=b2.
Solution
TheequationofthetangentatanypointPis
Theslopeofthetangentism.
Therefore,theslopeoftheperpendicularlineSYis Sisthepoint(ae,0).
∴TheequationofSYisy= (x−ae).
LetY,thefootoftheperpendicular,be(x1,y1)Thenfromequation(7.27),weget
Fromequation(7.28),weget
Addingequations(7.29)and(7.30),weget
Cancelling,
Thelocusof(x1,y1)isx2+y2=a2.Similarly,wecanprovethatthelocusofY′isalsothiscircle.Hence,YandY′lieonthiscircle.
Note7.12.1:Thiscircleiscalledtheauxiliarycircle(x2+y2=a2).Thisisthecircledescribedonthemajoraxisasdiameter.
Example7.10
IfnormalatapointPontheellipse meetsthemajoraxisatGthen
provethat:
i. CG=e2CN,whereCisthecentreoftheellipseandNisthefootoftheperpendicularfromPtothemajoraxis.
ii. SG=eSPwhereSisthefocusoftheellipse.
Solution
i. LetPbethepoint(x1,y1).
Theequationofthenormalat(x1,y1)is
Whenthismeetsthex-axis,y=0.
ii.
Example7.11
Inanellipse,provethatthetangentandnormalatanypointParetheexternalandinternalbisectorsoftheangleSPS′whereSandS′arethefoci.
Solution
LetP(x1,y1)beanypointontheellipse.
LetthenormalatPmeetthex-axisatL.TheequationofthenormalatPis
Wheny=0,x=e2x1∴Lis(e2x1,0).
Fromequations(7.33)and(7.34),weget
Fromequations(7.33)and(7.34),weget
Therefore,thenormalPListheinternalbisectorof SincethetangentatPis
perpendiculartothenormalatP,thetangentPistheexternalbisector.
Example7.12
Findtheanglesubtendedbyafocalchordoftheellipse passing
throughanendoftheminoraxisatthecentreoftheellipse.
Solution
Theequationoftheellipseis
Theequationofthefocalchordis
ThecombinedequationofthelinesCBandCQisgotbyhomogenizationoftheequationoftheellipsewiththehelpofstraightline(7.37).
TheanglebetweenthelinesCBandCQisgivenby
SincetheangleBCQisobtuse,θ=tan−1
Example7.13
Abarofgivenlengthmoveswithitsextremitiesontwofixedstraightlinesat
Abarofgivenlengthmoveswithitsextremitiesontwofixedstraightlinesatrightangles.Provethatanypointoftheroddescribesanellipse.
Solution
LetOAandOBbethetwoperpendicularlinesandABbetherodoffixedlength.LetP(x1,y1)beanypointoftherod.LettherodbeinclinedatanangleθwithOX.
(i.e.)
TakePA=aandPB=b.Thenx=OQ=RP=bcosθ,y=QP=bsinθ,
Hence,
Therefore,thelocusofPisanellipse.
Example7.14
Theequation25(x2−6x+9)+16y2=400representsanellipse.Findthecentreandfocioftheellipse.Howshouldtheaxisbetransformedsothattheellipseis
representedbytheequation
Solution
25(x2−6x+9)+16y2=40025(x−3)2+16y2=400
Dividingby400, Takex−3=X,y=Y.
Then
ThemajoraxisofthisellipseistheY-axis.
Centreis(3,0).Fociare(3,±ae)(i.e.) (i.e.)(3,±3).Nowshiftorigin
tothepoint(3,0)andthenrotatetheaxesthroughrightangles.Thenthe
equationoftheellipsebecomes
Example7.15
Showthatifs,s′arethelengthsoftheperpendicularonatangentfromthefoci,a,a′thosefromtheverlicesandethatfromthecentrethens,s′−e2=e2(aa′−c2)whereeistheeccentricity.
Solution
Lettheequationoftheellipsebe
FociareS(ae,0)andS′(−ae,0).VerticesareA(a,0)andA′(−a,0),centreis(0,
0).Anytangenttotheellipse(7.38)is
TheperpendiculardistancefromS(ae,0)is=
Fromequations(7.39)and(7.40),wegetss′−c2=e2(aa′−c2).
Example7.16
Acircleofradiusrisconcentricwiththeellipse .Provethateach
commontangentisinclinedtotheaxisatanangletan−1 andtowardsits
length.
Solution
Theequationoftheellipseis
Theequationofthecircleconcentricwiththeellipseis
Anytangenttotheellipseis
Anytangenttothecircleis
Ifthetangentisacommontangentthen
Therefore,theinclinationtothemajoraxisisθ=tan−1
Example7.17
Provethatthesumofthesquaresoftheperpendicularsofanytangentofan
ellipse fromtwopointsontheminoraxis,eachdistance from
thecentreis2a2.
Solution
Theequationoftheellipseis .
Anytangenttotheellipseis .Theperpendiculardistancefrom
tothetangentis
Theperpendiculardistancefrom isgivenby
Example7.18
Letdbetheperpendiculardistancefromthecentreoftheellipse tothe
tangentdrawnatapointPontheellipse.IfF1andF2arethetwofociofthe
ellipsethenshowthat .
Solution
Theequationoftheellipseis .LetP(x1,y1)beanypointonit.
Theequationofthetangentat(x1,y1)is .
TheperpendiculardistancefromConthistangentis
WeknowthatPF1=a−ex1,PF2=a+ex1
Fromequations(7.45)and(7.46),weget
Example7.19
Showthatthelocusofthemiddlepointsoftheportionofatangenttotheellipse
includedbetweentheaxesisthecurve
Solution
Anytangenttotheellipse is
Whenthetangentmeetsthex-axis,y=0.
Whenitmeetsthey-axis,x=0.
Therefore,thepointsofintersectionofthetangentswiththeaxesare
and Let(x1,y1)bethemidpointoflineAB.
Therefore,thelocusofP(x1,y1)is
Example7.20
Provethatthetangenttotheellipse meetstheellipse in
points,tangentsatwhichareatrightangles.
Solution
Anytangenttotheellipse is
AtQandRletthetangentsmeettheellipse
LetL(x1,y1)bethepointofintersectionoftangentsatQandR.ThenQRisthechordofcontactformL.Itsequationis
Equations(7.48)and(7.49)representthesameline.Identifyingequations(7.48)
and(7.49),weget
Therefore, Thelocusof(x1,y1)istheequationofthedirector
circleoftheellipse(7.49).However,directorcircleistheintersectionofperpendiculartangents.Hence,thetangentsatQandRareatrightangles.
Example7.21
AchordPQofanellipsesubtendsarightangleatthecentreoftheellipse
ShowthatthelocusoftheintersectionofthetangentsatQandRisthe
ellipse
Solution
Theequationofellipseis
LetR(x1,y1)bethepointofintersectionoftangentsatPandQ.TheequationofthechordofcontactofPQis
ThecombinedequationofCPandCQisgotbyhomogenizationofequation(7.51)withthehelpofequation(7.52).
Since ,coefficientofx2+coefficientofy2=0.
ThelocusofP(x1,y1)is
Example7.22
Showthatthelocusofpoleswithrespecttotheellipse ofanytangentto
theauxiliarycircleis
Solution
Let(x1,y1)bethepolewithrespecttotheellipse .
Thepolarof(x1,y1)is
Thisisatangenttotheauxiliarycirclex2+y2=a2.Theconditionforthatisc2=a2(1+m2).
Dividingby
Thelocusof(x1,y1)is
Example7.23
Showthatthelocusofpolesoftangentstothecircle(x−h)2+(y−k)2=r2with
respecttotheellipse is
Solution
Let(x1,y1)bethepolewithrespecttotheellipse .Thenthepolarof(x1,
y1)is Thislineisatangenttothecircle(x−h)2+(y−k)2=r2.The
conditionforthisisthattheradiusofthecircleshouldbeequaltotheperpendiculardistancefromthecentreonthetangents.
Therefore,thelocusof(x1,y1)is
Example7.24
Findthelocusofthepoleswithrespecttotheellipseofthetangentstotheparabolay2=4px.
Solution
Let(x1,y1)bethepolewithrespecttotheellipse Thepolarof(x1,y1)is
Thisisatangenttotheparabolay2=4px.
∴Theconditionis
Therefore,thelocusof(x1,y1)isa2py2+b4x=0.
Example7.25
AnytangenttoanellipseiscutbythetangentsattheextremitiesofthemajoraxisinthepointTandT′.ProvethatthecircledrawnonTT′asdiameterpassesthroughthefoci.
Solution
Lettheequationoftheellipsebe
TheendsofmajoraxisareA(a,0)andA′(−a,0).Anytangenttotheellipseis
ThistangentmeetsthetangentsatA,A′atTandT′,respectively.Thenthe
coordinatesofTandT′areT T′ Theequation
ofthecircleonTT′asdiameteris
ThiscirclepassesthroughthefociS(ae,0)andS′(−ae,0).
Example7.26
TheordinateNPofapointPontheellipseisproducedtomeetthetangentatoneendofthelatusrectumthroughthefocusSinQ.ProvethatQN=SP.
Solution
LetLSL′bethelatusrectumthroughthefocusS.TheequationoftangentatLis
LetPbethepoint(x1,y1).TheequationoftheordinateatPis
WhenthetangentatLmeetstheordinateatPinQ,thecoordinatesofQaregivenbysolvingequations(7.56)and(7.57).
or
y1=a−ex1∴QN=a−ex1
WeknowthatSP=a−ex1.Therefore,QN=SP.
Example7.27
ThetangentandnormalatapointPontheellipsemeettheminoraxisinTandQ.ProvethatTQsubtendsarightangleateachofthefoci.
Solution
Theequationofellipseis
TheequationofthetangentandnormalatP(x1,y1)is
WhenthetangentandnormalmeetintheminoraxisinTandQ,respectively,
thecoordinatesofTandQareT and
ThecoordinatesofSare(ae,0).
Slopeof
Slopeof
Now,m1m2=−1.Therefore,TQsubtendsarightangleatthefocusS.ThecoordinatesofS′are
(−ae,0).HenceitisprovedthatTQsubtendsarightangleatS′.
Example7.28
IfSandS′bethefocioftheellipse andebeitseccentricitythenprove
thattan wherePisanypointontheellipse.
Solution
Theequationofellipseis .
ThecoordinatesofSare(ae,0)andS′are(−ae,0).
∴SS′=2ae
InanyΔABC,weknowthattan wheresisthesemiperimeterof
ΔABC.
Let
Then
Hence,
Example7.29
AvariablepointPontheellipseofeccentricityeisjoinedtoitsfociSandS′.ProvethatthelocusoftheincentreoftheΔPSS′isanellipsewhoseeccentricity
is
Solution
Lettheequationoftheellipsebe .
ThecoordinatesofthefociareS(ae,0)andS′(−ae,0).LetP(h,k)beanypointontheellipse.ThenSP+S′P=2a.AlsoSS′=2ae.AlsoSP=a−eh,S′P=a+ek.Letthecoordinatesoftheincentrebe(x1,y1).Then
Since(h,k)liesontheellipse
ThelocusofP(x1,y1)is whichisanellipsewhoseeccentricitye1
isgivenby,
Therefore,thelocusoftheincentreoftheΔPSS′isanellipsewhoseeccentricity
e1is
Exercises
1. Findtheequationofthetangenttotheellipsewhichmakesequalinterceptsontheaxes.
Ans.:
2. Findthelengthoflatusrectum,eccentricity,equationofthedirectrixandfocioftheellipse25x2+
16y2=400.
Ans.:
3. Theequationtotheellipseis2x2+y2−8x−2y+1=0.Findthelengthofitssemiaxescoordinatesofthefoci,lengthoflatusrectumandequationofthedirectrix.
Ans.:2,2 ,(2,–1),(2,3),2 ,x−2=0,y−1=0
4. Provethat touchestheellipse andfindthecoordinatesofthepointof
contact.
Ans.:
5. IfpbethelengthoftheperpendicularfromthefocusSoftheellipse onthetangentsat
Pthenshowthat
6. IfSTbetheperpendicularfromthefocusSonthetangentatanypointPontheellipse
thenshowthatTliesontheauxiliarycircleoftheellipse.
7. Thelinexcosα+ysinα=pinterceptedbytheellipse subtendsarightangleatits
centreprovethatthevalueofpis
8. Ifthechordofcontactofthetangentsdrawnfromthepoint(α,β)totheellipse
touchesthecirclex2+y2=c2provethatthepoint(α,β)liesontheellipse
9. Pisapointontheellipse andQ,thecorrespondingpointontheauxiliarycircle.Ifthe
tangentatPtotheellipsecutstheminoraxisinT,thenprovethatthelineQTtouchestheauxiliarycircle.
10. Tangentstotheellipse makeanglesθ1andθ2withthemajoraxis.Findtheequation
ofthelocusoftheirintersectionwhentan(θ1+θ2)isaconstant.11. Showthatthelocusofthepointofintersectionoftwoperpendiculartangentstoanellipseisa
circle.12. Provethatachordofanellipseisdividedharmonicallybyanypointonitanditspolewithrespect
totheellipse.13. IfthepolarofPwithrespecttoanellipsepassesthroughthepointQ,showthatpolarofQpasses
throughP.14. Findtheconditionforthepoleofthestraightlinelx+my=1withrespecttotheellipse
maylieontheellipse
Ans.:a2l2+b2m2=4
15. Chordsoftheellipse touchthecirclex2+y2=r2.Findthelocusoftheirpoles.
16. Chordsoftheellipse alwaystouchtheellipse .Showthatthelocusofthe
polesis
17. ProvethattheperpendicularfromthefocusofanellipsewhosecentreisConanypolarofPwillmeetCPonthedirectrix.
18. Showthatthefocusofanellipseisthepoleofthecorrespondingdirectrix.
19. Atangenttotheellipse meetstheellipse atQandR.Showthatthe
locusofthepoleofQRwithrespecttothelatterisx2+y2=a2+b2.20. Ifthemidpointofachordliesonafixedlinelx+my+n=0,showthatthelocusofpoleofthe
chordistheellipse
21. Findthelocusofthepolesofchordsoftheellipse whichtouchtheparabolaay2=
−2b2x.
22. Theperpendicularfromthecentreoftheellipse onthepolarofapointwithrespectto
theellipseisequaltoc.Provethatthelocusofthepointistheellipse,
23. Showthatthelocusofthepoleswithrespecttoanellipseofastraightlinewhichtouchesthecircledescribedontheminoraxisoftheellipseasdiameter.
24. Showthatthelocusofpolesoftangentstotheellipse withrespecttox2+y2=abis
anequalellipse.25. Provethatthetangentsattheextremitiesoflatusrectumofanellipseintersectonthe
correspondingdirectrix.
26. Findthecoordinatesofallpointsofintersectionoftheellipse andthecirclex2+y2=
6.Writedowntheequationofthetangentstotheellipseandcircleatthepointofintersectionandfindtheanglebetweenthem.
27. Tangentsaredrawnfromanypointontheellipse tothecirclex2+y2=a2.Provethat
theirchordofcontacttouchestheellipsea2x2+b2y2=r4.
28. Provethattheanglebetweenthetangentstotheellipse andthecirclex2+y2=abat
theirpointofintersectionistan−1
29. Provethatthesumofthereciprocalsofthesquaresofanytwodiametersofanellipsewhichareatrightanglestooneanotherisaconstant.
30. Anellipseslidesbetweentwostraightlinesatrightanglestoeachother.Showthatthelocusofitscentreisacircle.
31. Showthatthelocusofthefootofperpendicularsdrawnfromthecentreoftheellipse
onanytangenttoitis(x2+y2)2=a2x2+b2y2.32. Twotangentstoanellipseinterestatrightangles.Provethatthesumofthesquaresofthechords
whichtheauxiliarycircleinterceptsonthemisconstantandequaltothesquareofthelinejoiningthefoci.
33. Showthattheconjugatelinesthroughafocusofanellipseareatrightangles.34. Anarchwayisintheformofasemi-ellipse,themajoraxisofwhichcoincideswiththeroadlevel.
Ifthebreadthoftheroadis34feetandamanwhois6feethigh,justreachesthetopwhen2feetfromasideoftheroad,findthegreatestheightofthearch.
35. IfthepoleofthenormaltoanellipseatPliesonthenormalatQthenshowthatthepoleofthenormalatQliesonthenormalatP.
36. PQ,PRisapairofperpendiculartangentstotheellipse .ProvethatQRalwaystouches
theellipse
37. Showthatthepoints(xr,yr),r=1,2,3arecollineariftheirpolarswithrespecttotheellipse
areconcurrent.
38. Ifl1andl2bethelengthoftwotangentstotheellipse atrightanglestooneanother,
provethat
39. IfRPandRQaretangentsfromanexternalpointR(x1,y1)totheellipse andSbethe
focusthenshowthat
7.13ECCENTRICANGLE
LetPbeapointontheellipseandP′bethecorrespondingpointontheauxiliarycircle.TheangleCP′makeswiththepositivedirectionofx-axisiscalledtheeccentricangleofthepointPontheellipse.Ifthisangleisdenotedbyθ,thenCN=acosθandNP′=asinθ.
Weknowthat
ThenthecoordinatesofanypointPare(CN,NP).
(i.e.)(acosθ,bsinθ)∴‘θ’iscalledtheeccentricangleanditisalsocalledtheparameterofthepointP.
7.14EQUATIONOFTHECHORDJOININGTHEPOINTS
Tofindtheequationofthechordjoiningthepointswhoseeccentricanglesare‘θ’and‘ϕ’:Thetwogivenpointsare(acosθ,bsinθ)and(acosϕ,bsinϕ).Theequation
ofthechordjoiningthetwopointsis
Dividingbyab,
Therefore,theequationofthechordjoiningthepointswhoseeccentricangles
are‘θ’‘ϕ’is
Note7.14.1:Thischordbecomesthetangentat‘θ’ifϕ=θ
∴Theequationofthetangentat‘θ’is
7.15EQUATIONOFTANGENTAT‘Θ’ONTHEELLIPSE
Theequationoftheellipseis .
Differentiatingwithrespecttox,weget,
Theequationofthetangentat‘θ’is,
Dividingbyab,
Theslopeofthenormalat
Therefore,theequationofthenormalatθis:
Dividingbysinθcosθ,weget,
Therefore,equationofnormalat‘θ’ontheellipse is
7.16CONORMALPOINTS
Ingeneral,fournormalscanbedrawnfromagivenpointtoanellipse.Ifα,β,γ,andδbetheeccentricanglesofthesefourconormalpointsthenα+β+γ+δisanoddmultipleofπ.Let(h,k)beagivenpoint.LetP(acosθ,bsinθ)beanypointontheellipse
.
Theequationofthenormalatθis
Ifthenormalpassesthrough(h,k)then
Thisisafourthdegreeequationintandhencetherearefourvaluesfort.Foreachvalueoft,thereisavalueofθandhencetherearefourvaluesofθsayα,β,γ,andδ.Hence,therearefournormalsfromagivenpointtoanellipse.
Hence, aretherootsoftheequation(7.63).
7.17CONCYCLICPOINTS
Acircleandanellipsewillcutfourpointsandthatthesumoftheeccentricanglesofthefourpointsofintersectionisanevenmultipleofπ.Lettheequationoftheellipsebe
Lettheequationofthecirclebe
Anypointontheellipseis(acosb,asinθ).Whenthecircleandtheellipseintersect,thispointliesonthecircle.
Substitutingthesevaluesinequation(7.67),weget
Equation(7.68)isafourthdegreeequationintandhencetherearefourvaluesfort,realorimaginary.Foreachvalueofttherecorrespondsavalueofθ.Henceingeneraltherearefourpointsofintersectionofacircleandanellipsewitheccentricanglesθ1,θ2,θ3,andθ4.Weknowthat,
7.18EQUATIONOFACHORDINTERMSOFITSMIDDLEPOINT
Tofindtheequationofachordintermofitsmiddlepoint:
Lettheequationoftheellipsebe
LetR(x1,y1)bethemidpointofachordPQofthisellipse.LettheequationofchordPQbe
Anypointonthislineis(x1+rcosθ,y1+rsinθ).Whenthechordmeetstheellipsethispointliesontheellipse(7.69).
IfR(x1,y1)isthemidpointofthechordPQthenthetwovaluesofrarethedistancesPRandRQwhichareequalinmagnitudebutoppositeinsign.
Theconditionforthisisthecoefficientofr=0.
Substituting inequation(7.72),weget
Hence,theequationofchordintermsofitsmiddlepointisT=S1
where
7.19COMBINEDEQUATIONOFPAIROFTANGENTS
Tofindthecombinedequationofpairoftangentsfrom(x1,y1)totheellipse
Lettheequationofthechordthrough(x1,y1)be
Anypointonthislineis(x1+rcosθ,y1+rsinθ)
Ifthispointliesontheellipse ,
Thetwovaluesofrarethedistancesofthepointofintersectionofthechordandtheellipsefrom(x1,y1).Thelinewillbecomeatangentifthetwovaluesofrareequal.Theconditionforthisisthediscriminantofthequadraticequationiszero.
Usingthevaluesofcosθandsinθfromequation(7.70a),
Thisisthecombinedequationofthepairoftangentsfrom(x1,y1).
Note7.19.1:Thecombinedequationofthepairoftangentsfromthepoint(x1,y1)is
Ifthetwotangentsareperpendicularthencoefficientofx2+coefficientofy2=0.
Thelocusof(x1,y1)isx2+y2=a2+b2.Therefore,thelocusofthepointofintersectionofperpendiculartangentsisacircle.Thisequationiscalledthedirectrixofthecircle.
7.20CONJUGATEDIAMETERS
Example7.30
Findtheconditionthatthelinelx+my+n=0maybeatangenttotheellipse
.
Solution
Letlx+my+n=0beatangenttotheellipse .
Letthelinebetangentat‘θ’.Theequationofthetangentatθis
However,theequationoftangentisgivenas
Identifyingequations(7.71a)and(7.72a),weget
Squaringandadding,weget
Thisistherequiredcondition.
Example7.31
Findtheconditionforthelinelx+my+n=0tobeanormaltotheellipse
.
Solution
Theequationoftheellipseis .
Theequationofnormalis
Letthisequationbenormalat‘θ’.Theequationofthenormalat‘θ’is
Theequations(7.73)and(7.74)representthesameline.Therefore,identifyingequations(7.73)and(7.74),weget
Squaringandaddingequations(7.78)and(7.79),weget
Thisisrequiredcondition.
Example7.32
Showthatthelocusofthepointofintersectionoftangentstoanellipseatthepointswhoseeccentricanglesdifferbyaconstantisanellipse.
Solution
LettheeccentricanglesofPandQbeα+βandα−β.∴(α+β)−(α−β)=2β=2k;aconstant∴β=k.
TheequationoftangentsatPandQare
Let(x1,y1)betheirpointofintersection.Then
Solvingequations(7.77)and(7.78),weget
Squaringandadding,weget
Sinceβ=k,thelocusof(x1,y1)is whichisanellipse.
Example7.33
Showthatthelocusofpolesofnormalchordsoftheellipse is
Solution
Let(x1,y1)bethepoleofthenormalchordoftheellipse
Thenthepolarof(x1,y1)withrespecttoellipseis
Letthisbenormalat‘θ’ontheellipseofequation(7.84).Thentheequationofthenormalat‘θ’is
Equations(7.80)and(7.81)representthesameline.Therefore,identifyingequations(7.80)and(7.81),weget
Squaringandaddingweget,
Therefore,thelocusof(x1,y1)is
Example7.34
Findthelocusofmidpointsofthenormalchordsoftheellipse .
Solution
Let(x1,y1)bethemidpointofachordoftheellipsewhichisnormalatθ.Theequationofthechordintermsofitsmiddlepointis
Theequationofthenormalat‘θ’is
Equations(7.82)and(7.83)representthesameline.Therefore,identifyingequations(7.87)and(7.88)weget,
Squaringandadding,weget
Squaringandadding,weget
Therefore,thelocusof(x1,y1)is
Example7.35
Ifthechordjoiningtwopoints,whoseeccentricanglesareαandβontheellipse
cutsthemajoraxisatadistancedfromthecentre,showthattan
Solution
Theequationofthechordjoiningthepointswhoseeccentricanglesareαand
Thislinemeetsthemajoraxisatthepoint(d,0).
Example7.36
Thetangentatthepointαontheellipsemeetauxiliarycircleontwopointswhichsubtendarightangleatthecentre.Showthattheeccentricityoftheellipseis(1+sin2α)–1/2.
Solution
Lettheequationoftheellipsebe
Theequationoftheauxiliarycircleis
Theequationofthetangentat .Thislinemeetstheauxiliary
circleatPandQ.ThenthecombinedequationofthelinesCQand
(i.e.)
since coefficientofx2+coefficientofy2=0
Example7.37
Ifthenormalattheendofalatusrectumofanellipsepassesthroughoneextremityoftheminoraxis,showthattheeccentricityofthecurveisgivenbye4
+e2−1=0.
Solution
Lettheequationoftheellipsebe .
ThecoordinatesoftheendLofthelatusrectumare Theequationofthe
normalatLis
Thisnormalpassesatthepoint
Thislinepassesthroughthepoint(0,−b).
Example7.38
Provethatthetangentandnormalatapointontheellipsebisecttheanglebetweenthefocalradiiofthatpoint.
Solution
Lettheequationoftheellipsebe
LetPTandPQbethetangentandnormalatanypointPontheellipse.The
equationofthenormalat(x1,y1)is
Whenthismeetthemajoraxis,y=0.
SinceSP′=a+ex1andSP=a−ex1.
Hence,PGbisectsinternally SincethetangentPTisperpendiculartoSG,
PTistheexternalbisectorof Therefore,thetangentandnormalatParethe
bisectorsoftheanglesbetweenthefocalradiithroughthatpoint.
Example7.39
Showthatthelocusofthemiddlepointofchordoftheellipse which
subtendsarightangleatthecentreis
Solution
Theequationoftheellipseis
Let(x1,y1)bethemidpointofachordoftheellipseofequation(7.87).
Thenitsequationis
IfCisthecentreoftheellipse,thecombinedequationofthelinesCPandCQis
Since ,coefficientofx2+coefficientofy2=0.
Example7.40
Provethattheportionofthetangenttotheellipseinterceptedbetweenthecurveandthedirectrixsubtendsarightangleatthecorrespondingfocus.
Solution
LetPbethepoint(acosθ,bsinθ)ontheellipse .Theequationofthe
tangentatθis
Theequationofthecorrespondingdirectrixis
Solvingequations(7.88)and(7.89),wegetT,thepointofintersection.
TheslopeofSPis
TheslopeofSTis
Example7.41
Anormalinclinedat45°tothex-axisoftheellipse isdrawn.Itmeets
themajorandminoraxisinPandQrespectively.IfCisthecentreoftheellipse,
showthattheareaof∆CPQis sq.units.
Solution
Theequationofthenormalat‘θ’is
Whenthismeetsx-axis,y=0.
Therefore,Pis
Whenitmeetsy-axis,x=0.
Therefore,Qis
Cis(0,0).
Slopeofthenormal=
Example7.42
Ifα−βisaconstant,provethatthechordjoiningthepoints,‘α’and‘β’touchesafixedellipse.
Solution
Theequationofthechordjoiningthepointsαandβis
Take thentheaboveequationbecomes =cosk.
Thislineisatangenttotheellipse
Example7.43
Ifthechordjoiningthevariablepointsatθandϕontheellipse
subtendsarightangleatthepoint(a,0)thenshowthat
Solution
Pisthepoint(acosθ,bsinθ).
Qisthepoint(acosϕ,bsinϕ).
SlopeofAPis
SlopeofAQis
SinceAPisperpendiculartoAQ,
Example7.44
Ifthenormaltotheellipse atthepointαcutsthecurvejoinin2αshow
thatcos
Solution
Theequationoftheellipseis
a2=14,b2=5
Theequationofthenormalat‘α’is
Example7.45
IfthenormalatanypointPtotheellipse meetsthemajorandminor
axesinGandgandifCFbetheperpendicularuponthisnormal,whereCisthecentreoftheellipse,thenprovethatPF·Pg=a2andPF·PG=b2.
Solution
LetP(acosθ,bsinθ)beanypointontheellipse.LetthenormalatPmeetthemajoraxisinGandminoraxising.LetCFbetheperpendicularfromCtothenormalatP.TheequationsofthetangentandnormalatPare
ThenthecoordinatesofGandgare and
PF=CLwhereCListheperpendicularonthetangent.
Example7.46
Showthattheconditionforthenormalsatthepoints(xi,yi),i=1,2,3onthe
ellipse tobeconcurrentis
Solution
Let(h,k)bethepointofconcurrenceofthenormal.Theequationofthenormal
at
Sincethisnormalpassesthrough(h,k),
Similarly,
Eliminatinghandkfromequations(7.94),(7.95)and(7.96),weget
Example7.47
Showthattheareaofthetriangleinscribedinanellipseis
aretheeccentricanglesofthevertices
andhencefindtheconditionthattheareaofthetriangleinscribedinanellipseismaximum.
Solution
LetΔABCbeinscribedintheellipse
LetA,BandCbethepoints(acosα,bsinα),(acosβ,bsinβ)and(acosγ,bsinγ),respectively.Thentheareaofthe∆ABCisgivenby,
IfA′,B′andC′arethecorrespondingpointsontheauxiliarycirclethen
AreaofΔABCisthegreatestwhentheareaofΔA′B′C′isthegreatest.However,theareaofA′B′C′isthegreatestwhenthetriangleisequilateral.In
thiscasetheeccentricanglesofthepointsP,QandRare
(i.e.)TheeccentricanglesofthepointsP,QandRdifferby
Example7.48
Ifthreeofthesidesofaquadrilateralinscribedinanellipseareinafixeddirection,showthatthefourthsideofthequadrilateralisalsoinafixeddirection.
Solution
Letα,β,γandδbetheeccentricanglesoftheverticesofthequadrilateralABCDinscribedintheellipse
ThentheequationofthechordPQis
TheslopeofthechordPQis .
SincethedirectionofPQisfixed, constant.
Similarly,
Therefore,thedirectionofPSisalsofixed.
Example7.49
Provethattheareaofthetriangleformedbythetangentsatthepointsα,βandγis
Solution
Theequationoftangentsatαandβare
Solvingequations(7.104)and(7.105)weget,
Therefore,thepointofintersectionoftangentsatPis
Hence,theareaofthetriangleformedbythetangentsatα,βandγis
Exercises
1. Ifαandβbetheeccentricanglesattheextremitiesofachordofanellipseofeccentricitye,prove
thatcos
2. LetPandQbetwopointsonthemajoraxisofanellipse equidistantfromthecentre.
ChordsaredrawnthroughPandQmeetingtheellipseatpointswhoseeccentricanglesareα,β,g
andδ.Thenprovethattan
3. Provethatthechordjoiningthepointsontheellipse whoseeccentricanglesdifferby
touchesanotherellipsewhosesemi-axesarehalfthoseofthefirst.
4. PSP′andQSQ′aretwofocalchordsofellipse suchthatPQisadiameter.Provethat
P′Q′passesthroughafixedpointonthemajoraxisoftheellipse.Findalsoitsequation.
Ans.:
5. PandP′arethecorrespondingpointsonanellipseanditsauxiliarycircle.ProvethatthetangentsatPandP′intersectonthemajoraxis.
6. ThetangentatoneendPofadiameterPP′ofanellipseandanychordP′QthroughtheotherendmeetatR.ProvethatthetangentatQbisectsPR.
7. Provethatthethreeellipses willhaveacommontangentif
8. AnytangenttotheellipseiscutbythetangentsattheendsofthemajoraxisinTandT′.ProvethatthecircleonTT′asdiameterwillpassthroughthefoci.
9. Findthecoordinatesofthepointsontheellipse ,thetangentsatwhichwillmakeequal
angleswiththeaxis.Alsoprovethatthelengthoftheperpendicularfromthecentreoneitherof
theseis
Ans.:
10. Findtheconditionforthelinexcosα+ysinα=pisatangenttotheellipse
Ans.:αcos2α+bsin2α=p2
11. Ifthetangenttotheellipse ,interceptslengthsαandβonthecoordinateaxesthen
showthat
12. Ifxcosα+ysinα−p=0beatangenttotheellipse ,provethatp2=a2cos2α+b2
sinα.IfPbethepointofcontactofthetangentxcosα+ysinα=pandN,thefootofthe
perpendicularonit,fromthecentreoftheellipse,provethat
13. ThetangentatoneendofPofadiameterOP′ofanellipseandanychordP′QthroughtheotherendmeetinR.ProvethatthetangentatQbisectsOR.
14. PandP′arecorrespondingpointsonanellipseandtheauxiliarycircle.ProvethatthetangentsatPandP′intersectonthemajoraxis.
15. IfthenormalatapointPontheellipseofsemi-axesa,bandcentreCcutsthemajorandminor
axesatGandg,showthata2Cg2+b2CG2=(a2−b2)2.AlsoprovethatPG=e·GN,wherePNistheordinateofP.
16. ThetangentsandnormalatapointPontheellipse meetthemajoraxisinTandT′so
thatTT′=a.ProvethattheeccentricangleofPisgivenbye2cos2θ+cosθ−1=0.17. Provethat,thelinejoiningtheextremitiesofanytwoperpendiculardiametersofanellipsealways
touchesaconcentriccircle.18. Showthatthelocusofthefootoftheperpendiculardrawnfromthecentreoftheellipse
onanytangenttoitis(x2+y2)2=(a2x2+b2y2)2.
19. IfPisanypointontheellipse whoseordinateisy′,provethattheanglebetweenthe
tangentatPandthefixeddistanceofPis
20. Showthatthefeetofthenormalsthatcanbedrawnfromthepoint(h,k)totheellipse
lieonthecurveb2(k−y)+a2y(x−h)=0.
21. Ifthenormalsatthefourpoints(xi,yi),i=1,2,3,4ontheellipse areconcurrent
showthat:
22. Ifthenormalsatthefourpointsθi,i=1,2,3,4areconcurrent,provethat(Σcosθi)(Σsecθi)=4.
Showthatthemeanpositionofthesefourpointsis where(h,k)isthepoint
ofconcurrency.
23. Ifthenormalsatthepointsα,βandγontheellipse areconcurrentthenprovethat
24. Ifα,β,γandδaretheeccentricanglesofthefourcornerpointsontheellipse then
provethat:(i)Σcos(α+β)=0and(ii)Σsin(α+β)=0.25. IfthepoleofthenormaltoanellipseatPliesonthenormalatQ,showthatthepoleofthenormal
atQliesonthenormalatP.26. FindthelocusofthemiddlepointsofthechordsofellipsewhosedistancefromthecentreCis
constantc.
27. Findthelocusofthemidpointofchordsoftheellipseofconstantlength2l.
28. Showthatthelocusofmidpointsofchordsoftheellipse ,tangentsattheendsof
whichintersectonthecirclex2+y2=a2is
29. Ifthemidpointofachordliesonafixedlinelx+my+n=0showthatthelocusofthepoleofthe
chordistheellipse
30. Showthatthelocusofmiddlepointsofthechordsoftheellipsethatpassthroughafixedpoint(h,
k)istheellipse
31. Provethatthelocusofthepointofintersectionoftangentstoanellipseattwopointswhoseeccentricanglesdifferbyaconstantisanellipse.Ifthesumoftheeccentricanglesbeconstantthenprovethatthelocusisastraightline.
32. TPandTQarethetangentsdrawntoanellipsefromapointTandCisitscentre.Provethatthe
areaofthequadrilateralCPTQisabtan whereθandϕaretheeccentricanglesofPandQ.
33. TheeccentricanglesoftwopointsPandQontheellipse areθandϕ.Provethatthe
areaofthisparallelogramformedbythetangentsattheendsofthediametersthroughPandQis4abcosec(θ−ϕ).
34. Chordsoftheellipse passthroughafixedpoint(h,k).Showthatthelocusoftheir
middlepointsistheellipse
35. IfPisanypointonthedirectorcircle,showthatthelocusofthemiddlepointsofthechordin
whichthepolarofPcutstheellipseis
36. Showthatthelocusofmidpointsofthechordsoftheellipse touchingtheellipse
37. IfthenormalstoanellipseatPi,i=1,2,3,4areconcurrentthenthecirclethroughP1,P2andP3meetstheellipseagaininapointP4whichistheotherendofthediameterthroughP4.
38. Findthecentreofthecirclepassingthroughthethreepoints,ontheellipsewhoseeccentricanglesareα,βandγ.
39. IfABCbeamaximumtriangleinscribedinanellipsethenshowthattheeccentricanglesofthe
verticesdifferby andthenormalsA,BandCareconcurrent.
40. Thetangentandnormaltotheellipsex2+4y2=2,atthepointPmeetthemajoraxisinQandR,
respectivelyandQR=2.ShowthattheeccentricangleofPiscos−1
41. Iftwoconcentricellipsesbesuchthatthefociofonelieontheotherthenprovethattheangle
betweentheiraxesis wheree1ande2aretheireccentricities.
42. Showthatthelengthofthefocalchordoftheellipse whichmakesanangleθwiththe
majoraxisis .
43. Ifthenormalsaredrawnattheextremitiesofafocalchordofanellipse,provethatalinethroughtheirpointofintersectionparalleltothemajoraxiswillbisectthechord.
44. Iftangentsfromthepointtotheellipse cutoffalengthequaltothemajoraxisfrom
thetangentat(a,0),provethatTliesonaparabola.45. IfthenormalatanypointPonanellipsecutsthemajoraxisatG,provethatthelocusofthe
middlepointofPQisanellipse.
46. Showthatthelocusoftheintersectionoftwonormalstotheellipse whichare
perpendiculartoeachotheris
47. Iftheanglebetweenthediameterofanypointoftheellipse andthenormalatthat
pointisθ,provethatthegreatestvalueof
48. Pisanypointonanellipse.ProvethatthelocusofthecentroidGofthepointPandthetwofocioftheellipseisaconcentricellipseofthesameeccentricity.
49. IfP,Q,RandSareconormalpointsonanellipse,showthatthecirclepassingthroughPandRwillcuttheellipseatapointS′whereSandS′aretheendsofadiameteroftheellipse.
50. Showthatthelocusofpoleofanytangenttotheellipsewithrespecttotheauxiliarycircleisasimilarconcentricellipsewhosemajoraxisisatrightanglestothatoftheoriginalellipse.
51. Thenormalsoffourpointsofanellipsemeetat(h,k).Iftwoofthepointslieon
provethattheothertwopointslieon
52. Ifthenormalstotheellipse attheendsofthechordslx+my=1andl1x+m1y=1be
concurrentthenshowthata2ll1=b2mm1=−1.
53. Provethattwostraightlinesthroughthepointsofintersectionofanellipsewithanycirclemakeequalangleswiththeaxesoftheellipse.
54. Showthattheequationofapairofstraightlineswhichareatrightanglesandeachofwhichpasses
throughthepoleoftheothermaybewrittenaslx+my+n=0andn(mx−ny)+lm(a2−b2)=0.Alsoprovethattheproductofthedistancesofsuchpairoflinesfromthecentrecommonly
exceeds
55. ShowthattherectangleundertheperpendiculardrawntothenormalatapointofanellipsefromthecentreandfromthepoleofthenormalisequaltotherectangleunderthefocaldistancesofP.
56. ProvethatifP,Q,RandSarethefeetofthenormalstotheellipse andthecoordinates
(x1,y1),(x2,y2),arethepolesofPQandRSthentheyareconnectedbytherelations
57. Ifthenormalsatfourpointsoftheellipse areconcurrentandiftwopointslieonthe
linelx+my=1,showthattheothertwopointslieontheline .Henceshowthatifthe
feetofthetwonormalsfromapointPtothisellipsearecoincidentthenthelocusofthemidpoints
ofthechordsjoiningthefeetoftheothernormalsis
7.20.1LocusofMidpoint
Locusofmidpointofaseriesofparallelchordsoftheellipse:Let(x1,y1)bethemidpointofachordparalleltotheliney=mx.ThentheequationofthechordisT=S1.
Itsslopeis
Sincethischordisparalleltoy=mx,
Thelocusof(x1,y1)is whichisastraightlinepassingthroughthecentre
oftheellipse.Ify=m1xbisectallchordsparalleltoy=mxthen
Bysymmetryofthisresult,weseethatthediametery=mxbisectallthechordsparalleltoy=m1x.
Definition7.20.1Twodiametersaresaidtobeconjugatetoeachotherifchordsparalleltooneisbisectedbytheother.Therefore,theconditionforthediameter
y=mxandy=m1xtobeconjugatediametersis
7.20.2Property:TheEccentricAnglesoftheExtremitiesofaPairofSemi-conjugateDiameterDifferbyaRightAngle
LetPCP′andDCD′beapairofconjugatediameters.LetPbethepoints(acosθ,
bsinθ)andDbethepoints(acosϕ,bsinϕ).ThentheslopeofCPis
TheslopeofCDis
SinceCPandCDaresemi-conjugatediameters
Therefore,theeccentricanglesofapairofsemi-conjugatediametersdifferbyarightangle.
Note7.20.1:ThecoordinatesofDare
(i.e.)(−asinθ,bcosθ)
Therefore,ifthecoordinatesofPare(acosθ,bsinθ)thenthecoordinatesofDare(−asinθ,bcosθ).ThecoordinatesofP′are(−acosθ,−bsinθ).ThecoordinatesofD′are(asinθ,−bcosθ).
7.20.3Property:IfCPandCDareaPairofSemi-conjugateDiametersthenCD2
+CP2isaConstant
ThecoordinatesofC,PandDareC(0,0)
P(acosθ,bsinθ)andD(−asinθ,bcosθ).
Then
7.20.4Property:TheTangentsattheExtremitiesofaPairofConjugateDiametersofanEllipseEnclosesaParallelogramWhoseAreaIsConstant
LetPCP′andDCD′beapairofconjugatediameters.LetPbethepoint(acosθ,
bsinθ).ThenDisthepoint
(i.e.)(−asinθ,bcosθ).
TheequationofthetangentatPis
TheslopeofthetangentatPis
TheslopeofCDis
Sincethetwoslopesareequal,thetangentsatPisparalleltoDCD′.Similarly,wecanshowthatthetangentatP′isparalleltoDCD′.Therefore,thetangentat
PandP′areparallel.Similarly,thetangentDandD′areparallel.Hence,thetangentsatP,P′,D,D′fromaparallelogramEFGH.TheareaoftheparallelogramEFGH
7.20.5Property:TheProductoftheFocalDistancesofaPointonanEllipseIsEqualtotheSquareoftheSemi-diameterWhichIsConjugatetotheDiameter
ThroughthePoint
LetSandS′bethefociofellipse .LetPbeanypointontheellipseand
drawMPM′perpendiculartothedirectrix.
Then,
7.20.6Property:IfPCP′andDCD′areConjugateDiameterthenTheyarealsoConjugateLines
Weknowthatthepolarofapointandthechordofcontactoftangentsfromittotheellipsearethesame.Therefore,thepoleofthediameterPCP′willbethepointofintersectionofthetangentsatPandP′whichareparallel.Therefore,thepoleofPCP′liesatinfinityontheconjugatediameterDCD′.Hence,PCP′andDCD′areconjugatelines.
Note7.20.2:Conjugatediameterisaspecialcaseofconjugatelines.
7.21EQUI-CONJUGATEDIAMETERS
Definition7.21.1Twodiametersofanellipsearesaidtobeequiconjugatediametersiftheyareofequallength.
7.21.1Property:Equi-conjugateDiametersofanEllipseLiealongtheDiagonalsoftheRectangleFormedbytheTangentattheEndsofitsAxes
LetPCP′andDCD′betwoconjugatediametersoftheellipse .Letthe
coordinatesofPbe(acosθ,bsinθ).ThenthecoordinatesofDare(−asinθ,bcosθ).CisthepointC(0,0).
IfCPandCDareequi-conjugatediametersthenCP2=CD2.
When theequationofthediameteris
When equationsofthesetwoconjugatediametersare Therefore,
theequi-conjugatediametersare whicharetheequationsofthe
diagonalsformedbythetangentsatthefourverticesoftheellipse.
ILLUSTRATIVEEXAMPLESBASEDONCONJUGATEDIAMETERS
Example7.50
Showthatthelocusofthepointofintersectionoftangentsattheextremitiesofa
pairofconjugatediametersoftheellipse istheellipse
Solution
LetPCP′andDCD′beapairofconjugatediametersoftheellipse .
LetPbethepoint(acosθbsinθ).ThenD′isthepoint
Let(x1,y1)bethepointofintersectionofthetangentsatPandD.TheequationsofthetangentsatPandDare
Sincethesetwotangentsmeetat(x1,y1),
and
Squaringandaddingfromequations(7.103)and(7.104),weget
Therefore,thelocusof(x1,y1)is
Example7.51
IfPandDaretheextremitiesofapairofconjugatediameteroftheellipse
showthatthelocusofthemidpointofPDis
Solution
LetPbethepoint(acosθ,bsinθ).ThenthecoordinatesofDare(−asinθ,bcosθ).Let(x1,y1)bethemidpointofPD.
Squaringandaddingfromequations(7.105)and(7.106),weget
Therefore,thelocusof(x1,y1)is whichisaconcentricellipse.
Example7.52
IfCPandCDaretwoconjugatesemi-diametersofanellipse then
provethatthelinePDtouchestheellipse
Solution
LettheeccentricangleofPbeθ.ThentheeccentricangleofDis The
equationofthechordPDis
(i.e)
where
Thisstraightlinetouchestheellipse
Example7.53
FindtheconditionthatthetwostraightlinesrepresentedbyAx2+2Hxy+By2=
0maybeapairofconjugatediametersoftheellipse .
Solution
LetthetwostraightlinesrepresentedbyAx2+2Hxy+By2=0bey=m1xandy
=m2x.Then
Theconditionforthelinestobeconjugatediametersis
Thisistherequiredcondition.
Example7.54
IfPandDbetheendsofconjugatesemi-diametersoftheellipsethenshowthatthelocusofthefootoftheperpendicularfromthecentreonthelinePDis2(x2+y2)2=a2x2+b2y2.
Solution
LettheeccentricangleofPbeθ.ThentheeccentricangleofDisθ+ .The
equationofPDis
Theequationofthelineperpendiculartothisandpassingthroughthecentre(0,0)is
Let(x1,y1)bethefootoftheperpendicularfrom(0,0)onPD.Then(x1,y1)liesontheabovetwolines.
Solvingfor and ,weget
Substitutingfor and inequation(7.107),weget
Therefore,thelocusof(x1,y1)is2(x2+y2)=a2x2+b2y2.
Example7.55
CPandCDaresemi-conjugatediametersoftheellipse .Ifthecircleson
CPandCDasdiametersintersectinRthenprovethatthelocusofthepointRis2(x2+y2)2=a2x2+b2y2.
Solution
LetPbethepoint(acosθ,bsinθ).ThenDisthepoint(−asinθ,bcosθ).Cisthepoint(0,0).TheequationsofthecirclesonCPandCDasdiametersarex(x−acosθ)+
y(y−bsinθ)=0andx(x+asinθ)+y(y−bcosθ)=0.
(i.e.)x2+y2=axcosθ+bysinθandx2+y2=−axsinθ+bycosθ.Let(x1,y1)beapointofintersectionofthesetwocircles.Then
Bysquaringandaddingequations(7.111)and(7.112),weget
Therefore,thelocusof(x1,y1)is(x2+y2)2=a2x2+b2y2.
Example7.56
Ifthepointsofintersectionoftheellipses and bethepointsof
conjugatediametersoftheformerprovethat
Solution
Anyconicpassingthroughthepointofintersectionoftheellipses
and
is
whereλ=−1,equation(7.118)reducesto
Thisbeingahomogeneousequationofseconddegreeinxandyrepresentsapairofstraightlines,thatis,equation(7.116)representsapairofstraightlines
passingthroughtheorigin.
or
Example7.57
Ifαandβbetheanglessubtendedbythemajoraxisofanellipseattheextremitiesofapairofconjugatediametersthenshowthatcos2α+cos2βisaconstant.
Solution
Letequationoftheellipsebe .
LetPbethepoint(acosα,bsinβ).ThenDisthepoint
TheslopeofAPis
TheslopeofA′Pis
Changingaintoα+ ,
Addingequations(7.117)and(7.118),weget
Example7.58
Ifxcosα+ysinα=pisachordjoiningtheendsPandDofconjugatesemi-diametersoftheellipsethenprovethata2cos2α+b2sin2α=2p2.
Solution
Lettheequationoftheellipsebe .LetPCP′andDCD′beapairof
conjugatediameters.LetPbethepoint(acosθ,bsinθ)thenDisthepoint
TheequationofPDis
However,theequationofPDisgivenas
Equations(7.119)and(7.120)representthesameline.Identifyingequations(7.119)and(7.120),weget
Example7.59
CPandCDareconjugatediametersoftheellipse .Atangentisdrawn
paralleltoPDmeetingCPandCDinRandSrespectively.ProvethatRandSlie
ontheellipse
Solution
LetCPandCDbeapairofconjugatediametersoftheellipse
LetPbethepoint(acosθ,bsinθ).ThenDisthepoint(−asinθ,bcosθ).SlopeofPDis
LettheequationofthetangentparalleltoPDbe
LetRbethepoint(h,k).Since(h,k)liesonthistangent,
Inaddition,theequationofCPis
Sincethispassesthrough(h,k),
Substitutinginequation(7.122),weget
Eliminatingmfromequations(7.123)and(7.124),
Dividingbya2b2, Thelocusof(h,k)is Similarly,thepointS
alsoliesontheaboveellipse.
Example7.60
Atangenttotheellipse cutsthecirclex2+y2=a2+b2inPandQ.
ProvethatCPandCQarealongconjugatessemi-diametersoftheellipsewhereCisthecentreofthecircle.
Solution
Theequationoftheellipseis
Theequationofthecircleisx2+y2=a2+b2.(7.125)Theequationofthetangentatθontheellipseis
ThismeetsthecircleinPandQ.ThecombinedequationCPandCQisgotbyhomogenizationofequation(7.125)withthehelpofequation(7.126),
∴CPandCQareconjugatesemi-diametersoftheellipse.
Example7.61
Provethattheacuteanglebetweentwoconjugatediametersisleastwhentheyareofequallength.
Solution
LetPCP′andDCD′betheconjugatediameters.
Fromequations(7.127)and(7.128),
RHSisleastwhenthedenominatoristhelargest.Thishappenswhen
Therefore,theacuteanglebetweenthediametersisminimumwhentheconjugatediametersareofequallengthandtheleastacuteangleisgivenby
Example7.62
Findthelocusofthepointofintersectionofnormalsattwopointsonanellipsewhichareextremitiesofconjugatediameters.
Solution
Lettheequationoftheellipsebe
Lettheequationoftheellipsebe
LetPandDbetheextremitiesofapairofconjugatediametersoftheellipse(7.129).LetPandDbethepointsP(acosθ,bsinθ)andD(−asinθ,bcosθ).TheequationsofthenormalatPandDare
Solvingequations(7.130)and(7.131),weget,
Squaringandadding,weget
Therefore,thelocusofthepointofintersectionofthesetwonormalsis(a2x2+b2y2)3=(a2−b2)2(a2x2−b2y2)2.
Example7.63
Ifthepointofintersectionoftheellipses and beatthe
extremitiesoftheconjugatediametersoftheformerthenprovethat
Solution
Thegivenellipsesare
Solvingequations(7.134)and(7.135)wegettheirpointofintersections.
Equation(7.134)−(7.135)gives
Thisisapairofstraightlinespassingthroughtheorigin.Ify=mxisoneofthe
linesthen
Thisisaquadraticequationinm.Ifm1andm2aretheslopesofthetwostraightlinesthroughtheoriginthen
Ifm1andm2aretheslopesofthepairofconjugatediametersthen
Fromequations(7.137)and(7.138),weget
Example7.64
LetPandQbetheextremitiesoftwoconjugatediametersoftheellipse
andSbethefocus.ThenprovethatPQ2−(SP−SQ)2=2b2.
Solution
LetSbe(ae,0)andPbe(acosθ,bsinθ).ThenSP=a−aecosθ,SQ=a+aesinθ.
Example7.65
IfCPandCDaresemi-conjugatediametersoftheellipse ,provethat
thelotusoftheorthocentreofΔPCDis2(b2y2+a2x2)3=(a2−b2)2(a2x2−b2y2)2.
Solution
LetPbethepoint(asinθ,bcosθ).ThenDis(−asinθ,bcosθ).TangentatPisparalleltoCD.TangentatDisparalleltoCP.Therefore,thealtitudesthroughPandDarethenormalsatPandD,
respectively.Let(x1,y1)betheorthocentre.TheequationofthenormalatPis
TheequationofthenormalatQis
Solvingequations(7.139)and(7.140)wegetthecoordinatesoftheorthocentre.
Squaringandadding,weget
Therefore,thelocusoftheorthocentreis2(a2x2+b2y2)3=(a2−b2)2(a2x2−b2y2)2.
Exercises
1. LetCPandCQbeapairofconjugatediametersofanellipseandletthetangentsatPandQmeetatR.ShowthatCRandPQbisecteachother.
2. Findtheconditionthatforthediametersof throughitspointsofintersectionwiththe
linelx+my+n=0tobeconjugate.
Ans.:l2+m2=a2l2+b2m2
3. Provethatb2x2+2hxy−a2y2=0representsconjugatediametersoftheellipse forall
valuesofh.
4. Provethata2x2+2hxy−b2y2=0representsconjugatediametersoftheellipseax2+by2=1forallvaluesofh.
5. Findthecoordinatesoftheendsofthediameteroftheellipse16x2+25y2=400whichisconjugateto5y=4x.
Ans.:
6. Findthelengthofsemi-diameterconjugatetothediameterwhoseequationisy=3x.7. Throughthefociofanellipse,perpendicularsaredrawntoapairofconjugatediameters.Prove
thattheymeetonaconcentricellipse.
8. Adiameteroftheellipse meetsonelatusrectuminPandtheconjugatediameter
meetstheotherlatusrectuminQ.ProvethatPQtouches
9. IfPP′isadiameterandQisanypointontheellipse,provethatQPandQP′areparalleltoapairofconjugatediametersoftheellipse.
10. Ifα+β=γ(aconstant)thenprovethatthetangentsataandbontheellipse .intersect
onthediameterthroughγ.11. Showthatthelinejoiningtheextremitiesofanytwodiametersofanellipsewhichareatright
anglestooneanotherwillalwaystouchafixedcircle.12. Showthatthesumofthereciprocalsofthesquareofanytwodiametersofanellipsewhichareat
rightanglestooneanotherisaconstant.
13. PandQareextremitiesoftwoconjugatediametersoftheellipse .andSisafocus.
ProvethatPQ2+(SP−SQ)2=2b2.14. Ifthedistancebetweenthetwofociofanellipsesubtendsangles2αand2βattheendsofapairof
conjugatediameters.Showthattan2α+tan2βisaconstant.15. Showthatthesumofthesquaresofthenormalattheextremitiesofconjugatesemi-diametersand
terminatedbymajoraxisisa2(1−e2)(2−e2).16. IfPandQaretwopointsonanellipsesuchthatCPisconjugatetothenormalatQ,provethatCQ
isconjugatetothenormalatP.
17. Twoconjugatediametersoftheellipse centreatCmeetthetangentatanypointPisE
andF.ProvethatPE·PF=CD2.
18. IfCPandCDareconjugatesemi-diametersoftheellipse ,thenormalatPcutsthe
majoraxisatGandthelineDCinFthenprovethatPG:CD=b:a.
19. ThenormalatavariablepointPofanellipse cutsthediameterCDconjugatetoPin
Q.ProvethattheequationofthelocusofQis
20. Showthatforaparallelograminscribedinanellipse,thesumofthesquaresofthesidesisconstant.
21. Showthatthemaximumvalueofthesmalleroftwoanglesbetweentwoconjugatediametersofan
ellipseis andtheminimumvalueofthisangleis whereaandbareitssemi-
majorandsemi-minoraxes,respectively.
22. IfPCP′andDCD′aretwoconjugatediametersoftheellipse .andQisanypointonthe
circlex2+y2=c2thenprovethatPQ2+DQ2+P2Q2+PQ2=2(a2+b2+2c2).
23. Twoconjugatediametersoftheellipse cutthecirclex2+y2=r2atPandQ.Show
thatthelocusofthemidpointofPQisa2[(x2+y2)2−r2x2]+b2[(x2+y2)2−r2y2]=0.24. Inanellipsewhosesemi-axesareaandb,provethattheacute-anglebetweentwoconjugate
diameterscannotbelessthan
25. IfCPandCDareconjugatediametersofanellipseshowthat4(CP2−CD2)=(SP−S′P)2−(SD
−S′D)2.26. Twoconjugatesemi-diametersofanellipseareinclinedatanglesαandβtothemajoraxis.Show
thattheirlengthscanddareconnectingtherelationc2sin2α+d2sin2β=0.27. Findtheconditionforthelinesl1x+m1y=0andl2x+m2y=0tobeconjugatediametersof
.
Ans.:a2ll1+b2mm1=0
28. Showthatax2+2hxy−by2=0representsconjugatediametersoftheellipseax2+by2=1forallvaluesofa.
29. Provethatax2+2hxy−by2=0representsconjugatediametersoftheellipseax2+by2=1forallvaluesofh.
30. CPandCQareconjugatesemi-diametersoftheellipse .AtangentparalleltoPQ
meetsCPandCQinRandS,respectively.ShowthatRandSlieontheellipse
31. IftwoconjugatediametersCPandCQofanellipsecutthedirectorcircleinLandM,provethatLMtouchestheellipse.
32. Twoconjugatediametersoftheellipse cutsthecirclex2+y2=ratPandQ.Show
thatthelocusofthemidpointofPQisa2[(x2+y2)2−r2x2]+b2[(x2+y2)2−r2y2]=0.
33. TheeccentricanglesoftwopointsPandQontheellipse areαandβ.Provethatthe
areaoftheparallelogramformedbythetangentsattheendsofthediametersthroughPandQis
andhenceshowthatitisleastwhenPandQaretheextremitiesofapairofconjugate
diameters.
34. LetPCP′beadiameteroftheellipse .IfthenormalatPmeetstheordinateatP′inT,
showthatthelocusofTis
35. IftwoconjugatediametersCPandCQofanellipsecutthedirectorcircleinLandM,provethatLMtouchestheellipse.
36. Inanellipse,apairofconjugatediametersisproducedtomeetadirectrix.Showthattheorthocentreofthetrianglessoformedisafocus.
37. ThroughafixedpointP,apairoflinesisdrawnparalleltoavariablepairofconjugatediametersofagivenellipse.ThelinesmeettheprincipalaxesinQandR,respectively.ShowthatthemidpointofQRliesonafixedline.
38. PerpendicularsPMandPNaredrawnfromanypointPofanellipseontheequi-conjugatediameteroftheellipse.ProvethattheperpendicularsfromPtoitspolarbisectMN.
39. Intheellipse3x2+7y2=21,findtheequationsoftheequi-conjugatediametersandtheirlengths.
Ans.:
40. Provethatthetangentstotheellipse atthepointswhoseeccentricanglesareθand
meetononeoftheequi-conjugatediameters.
41. Fromapointononeoftheequi-conjugatediametersofanellipsetangentsaredrawntotheellipse.
Showthatthesumoftheeccentricanglesofthepointofcontactisanoddmultipleof .
42. Tangentsaredrawnfromanypointontheellipse tothecirclex2+y2=r2.Provethat
thechordsofcontactaretangentstotheellipsea2x2+b2y2=r4.If ,provethatthe
lineandthecentretothepointsofcontactwiththecircleareconjugatediametersofthesecondellipse.
43. AnytangenttoanellipsemeetsthedirectorcircleinPandD.ProvethatCPandCDareinthedirectionsofconjugatediametersoftheellipse.
44. IfCPisconjugatetothenormalatQ,provethatCQisconjugatetothenormalatP.45. Provethatthestraightlinesjoiningthecentretotheintersectionofthestraightline
withtheellipseareconjugatediameters.
Chapter8
Hyperbola
8.1DEFINITION
Ahyperbolaisdefinedasthelocusofapointthatmovesinaplanesuchthatitsdistancefromafixedpointisalwaysetimes(e>1)itsdistancefromafixedline.Thefixedpointiscalledthefocusofthehyperbola.Thefixedstraightlineiscalledthedirectrixandtheconstanteiscalledtheeccentricityofthehyperbola.
8.2STANDARDEQUATION
LetSbethefocusandthelinelbethedirectrix.DrawSXperpendiculartothedirectrix.DivideSXinternallyandexternallyintheratioe:1(e>1).LetAand
A′bethepointofdivision.Since and thepointsAandA′lieonthe
curve.LetAA′=2aandCbeitsmiddlepoint.
Addingequations(8.1)and(8.2),weget
Subtractingequation(8.1)fromequation(8.2),weget
TakeCSasthex-axisandCYperpendiculartoCXasthey-axis.Then,thecoordinatesofSare(ae,0).LetP(x,y)beanypointonthecurve.DrawPMperpendiculartothedirectrixandPNperpendiculartox-axis.
Fromthefocusdirectrixpropertyofhyperbola,
Dividingbya2(e2−1),weget
Thisiscalledthestandardequationofhyperbola.
Note8.2.1:
1. Thecurvemeetsthex-axisatpoints(a,0)and(−a,0).
2. Whenx=0,y2=−a2.Therefore,thecurvemeetsthey-axisonlyatimaginarypoints,thatis,therearenorealpointsofintersectionofthecurveandy-axis.
3. If(x,y)isapointonthecurve,(x,−y)and(−x,y)arealsopointsonthecurve.Thisshowsthatthecurveissymmetricalaboutboththeaxes.
4. Foranyvalueofy,therearetwovaluesofx;asyincreases,xincreasesandwheny→∞,xalso→∞.Thecurveconsistsoftwosymmetricalbranches,eachextendingtoinfinityinboththedirections.
5. AA′iscalledthetransverseaxisanditslengthis2a.6. BB′iscalledtheconjugateaxisanditslengthis2b.
7. Ahyperbolainwhicha=biscalledarectangularhyperbola.Itsequationisx2−y2=a2.Its
eccentricityis
8. ThedoubleordinatethroughthefocusSiscalledlatusrectumanditslengthis
9. ThereisasecondfocusS′andaseconddirectrixl′tothehyperbola.
8.3IMPORTANTPROPERTYOFHYPERBOLA
Thedifferenceofthefocaldistancesofanypointonthehyperbolaisequaltothelengthoftransverseaxis.
8.4EQUATIONOFHYPERBOLAINPARAMETRICFORM
(asecθ,btanθ)isapointonthehyperbola forallvaluesofθ,θis
calledaparameterandisdenotedby‘θ’.Theparametricequationsofhyperbolaarex=asecθ,y=btanθ.
8.5RECTANGULARHYPERBOLA
Ahyperbolainwhichb=aiscalledarectangularhyperbola.Thestandardequationoftherectangularhyperbolaisx2−y2=a2.
8.6CONJUGATEHYPERBOLA
ThefociareS(ae,0)andS′(−ae,0)andtheequationsofthedirectricesare
Bythesymmetryofthehyperbola,ifwetakethetransverseaxisasthey-
axisandtheconjugateaxisasx-axis,thentheequationofthehyperbolais
Thishyperbolaiscalledtheconjugatehyperbola.Here,thecoordinatesofthe
fociareS(0,be)andS′(0,−be).Theequationsofthedirectricesare
Thelengthofthetransverseaxisis2b.Thelengthoftheconjugateaxisis2a.
Thelengthofthelatusrectumis
Thefollowingaresomeofthestandardresultsofthehyperbolawhose
equationis
1. Theequationofthetangentat(x1,y1)is
2. Theequationofthenormalat(x1,y1)is
3. Theequationofthechordofcontactoftangentsfrom(x1,y1)is
4. Thepolarof(x1,y1)is
5. Theconditionthatthestraightliney=mx+cisatangenttothehyperbolaisc2=a2m2−b2and
istheequationofatangent.
6. Theequationofthechordofthehyperbolahaving(x1,y1)asthemidpointis
7. Theequationofthepairoftangentsfrom(x1,y1)isT2=SS1
8. Parametricrepresentation:x=asecθ,y=btanθisapointonthehyperbolaandthispointisdenotedbyθ.θiscalledaparameterofthehyperbola.
Theequationofthetangentat
Theequationofthenormalat
9. Thecircledescribedonthetransverseaxisasdiameteriscalledtheauxiliarycircleandits
equationisx2+y2=a2.
10. Theequationofthedirectorcircle(thelocusofthepointofintersectionofperpendiculartangents)
isx2+y2=a2−b2.
Example8.6.1
Findtheequationofthehyperbolawhosefocusis(1,2),directrix2x+y=1and
eccentricity
Solution
LetP(x1,y1)beanypointonthehyperbola.Then
Hence,theequationofthehyperbolawhichisthelocusof
(x1,y1)is7x2+12xy−2y2−2x+14y−22=0.
Example8.6.2
Showthattheequationofthehyperbolahavingfocus(2,0),eccentricity2anddirectrixx−y=0isx2+y2−4xy+4x−4=0.
Solution
Sis(2,0):e=2andequationofthedirectrixisx−y=0.LetP(x,y)beany
pointonthehyperbola.Then,
Hence,theequationofthehyperbolaisx2+y2−4xy+4x−4=0.
Example8.6.3
Findtheequationofthehyperbolawhosefocusis(2,2),eccentricity and
directrix
3x−4y=1
Solution
Sis(2,2): anddirectrix3x−4y=1.LetP(x,y)beanypointonthe
hyperbola.
Hence,theequationofthehyperbolais19x2+216xy−44y2−346x−472y−791=0.
Example8.6.4
Findtheequationofthehyperbolawhosefocusis(0,0),eccentricity and
directrixxcosα+ysinα=p
Solution
Foranypointonthehyperbola,
Hence,theequationofthehyperbolais16(x2+y2)−25(xcosα+ysinα−p)2
=0.
Example8.6.5
Findtheequationofthehyperbolawhosefociare(6,4)and(−4,4)andeccentricity2.
Solution
Sis(6,4)andS′(−4,4),andCisthemidpointofSS′
Hence,theequationofthehyperbolais
Example8.6.6
Findtheequationofthehyperbolawhosecenteris(−3,2),oneendofthe
transverseaxisis(−3,4)andeccentricityis
Solution
Centreis(−3,4)Ais(−3,4)∴A′is(−3,6);a=2
Hence,theequationofthehyperbolais
(sincethelineparalleltoy-axisisthetransverseaxis)
Example8.6.7
Findtheequationofthehyperbolawhosecentreis(1,0),onefocusis(6,0),andlengthoftransverseaxisis6.
Solution
Hence,theequationofthehyperbolais (i.e.)16x2−9y2−32x−
128=0.
Example8.6.8
Findtheequationofthehyperbolawhosecentreis(3,2),onefocusis(5,2)andonevertexis(4,2).
Solution
Cis(3,2),Ais(4,2)andSis(5,2).Hence,CA=1andthetransverseaxisisparalleltox-axis.
∴a=1Alsoae=2.Sincea=1ande=2,b2=a2(e2−1)=1(4−1)=3.
Hence,theequationofthehyperbolais
Example8.6.9
Findtheequationofthehyperbolawhosecentreis(6,2),onefocusis(4,2)ande=2.
Solution
Transverseaxisisparalleltox-axisandCS=2unitsinmagnitude.
Hence,theequationofthehyperbolais
Example8.6.10
Findthecentre,eccentricityandfociofhyperbola9x2−16y2=144.
Solution
Dividingby144,weget
Hence,thecentreofthehyperbolais(0,0)
Hence,thefociare(5,0)and(−5,0).
Example8.6.11
Findthecentre,fociandeccentricityof12x2−4y2−24x+32y−127=0
Solution
Hence,centreis(1,4).
Hence,thefociare(6,4)and(−4,4).
Example8.6.12
Findthecentreandeccentricityofthehyperbola9x2−4y2+18x+16y−43=0.
Solution
Hence,centreis(−1,2),a2=4andb2=9.
Example8.6.13
IffromthecentreCofthehyperbolax2−y2=a2,CMisdrawnperpendiculartothetangentatanypointofthecurvemeetingthetangentatMandthecurveatN,showthatCM·CN=a2.
Solution
TheequationofthetangentatP(x1,y1)inx2−y2=a2isxx1−yy1=a2.
TheequationofthelineCNisxy1+yx1=0ThenCM,whichisperpendicularfromConthetangent,isgivenby
Solvingx2−y2=a2andxy1+yx1=0wegetthecoordinatesofN
Example8.6.14
Tangentstothehyperbolamakeanglesθ1,θ2withthetransverseaxis.Findtheequationofthelocusofpointofintersectionsuchthattan(θ1+θ2)isaconstant.
Solution
Lettheequationofthehyperbolabe .Then,theequationofthetangent
tothehyperbolais
Ifthistangentpassesthrough(x1,y1),then
Ifm1andm2aretheslopesofthetwotangents,then
Itisgiventhattan(θ1+θ2)=k.
Hence,thelocusof(x1,y1)isk(x2+y2−a2−b2)2xy=0.
Example8.6.15
Provethattwotangentsthatcanbedrawnfromanypointonthehyperbolax2−
y2=a2−b2totheellipse whichmakecomplementaryangleswiththe
axes.
Solution
Thetangentdrawnfromanypointtotheellipse is
Sincethispassesthrough(x1,y1)
Ifm1andm2aretheslopesofthetangents,then
Since(x1,y1)liesonx2−y2=a2−b2,wehave
Hence,thetwotangentsmakecomplementaryangleswiththeaxes.
Example8.6.16
Chordsofthehyperbola areataconstantdistancefromthecentre.Find
thelocusoftheirpoles.
Solution
Let(x1,y1)bethepolewithrespecttohyperbola .Thepolarof(x1,y1)
is Theperpendiculardistancefromthecentreonthepolaris
(aconstant)(i.e.)
Hence,thelocusof(x1,y1)is
Example8.6.17
Findtheequationofcommontangentstothehyperbolas and
Solution
Thetwogivenhyperbolasare and
Theconditionsfory=mx+ctobeatangenttothehyperbolasare
c2=a2m2−b2and
Hence,therearetwocommontangentswhoseequationsare
Example8.6.18
Showthatthelocusofmidpointsofnormalchordsofthehyperbolax2−y2=a2
is
(y2−x2)3=4a2x2y2.
Solution
Theequationofthehyperbolaisx2−y2=a2.Let(x1,y1)bethemidpointofanormalchordofthehyperbola.Theequation
ofthenormalis andtheequationofthechordintermsofthe
middlepointis Boththeseequationsrepresentthesameline.
Hence,identifyingthem,weget
Squaringandsubtracting,weget
Thefocusof(x1,y1)is(y2−x2)3=4a2x2y2.
Example8.6.19
Provethatthelocusofmiddlepointsofchordsofthehyperbola
passingthroughafixedpoint(h,k)isahyperbolawhosecentreis
Solution
Theequationofthehyperbolais .
Theequationofthechordofthehyperbolaintermsofitsmiddlepointis
Sincethischordpassesthroughthefixedpoint(h,k),
Thelocusof(x1,y1)is ,whichisahyperbolawhose
centreis
Example8.6.20
Showthatthelocusofthefootoftheperpendicularfromthecentreuponany
normaltothehyperbola
Solution
LetP(asecθ,btanθ)beapointonthehyperbola.Letm(x1,y1)bethefootoftheperpendicularfromthecentrewithnormalatP.Theequationofthenormal
atPis
TheequationoftheperpendicularfromC(0,0)onthisnormalis
Thesetwolinesintersectat(x1,y1)
Solvingthesetwoequationforx1andy1weget
Hence,thelocusof(x1,y1)is
Example8.6.21
Chordsofthecurvex2+y2=a2touchthehyperbola .Provethattheir
middlepointslieonthecurve(x2+y2)2=a2x2−b2y2.
Solution
Let(x1,y1)bethemidpointofthechordofthecircle.Itsequationis
Thisisatangenttothehyperbola .
Hence,theconditionis
Hence,thelocusof(x1,y1)is(x2+y2)2=(a2x2−b2y2).
Example8.6.22
Showthatthelocusofmidpointsofnormalchordsofthehyperbolax2−y2=a2
is(y2−x2)2=4a2xy.
Solution
Let(x1,y1)bethemidpointofthenormalchordofthehyperbolax2−y2=a2.Then,theequationofthechordis
Theequationofthenormalat‘θ’is
Thesetwoequationsrepresentthesameline.Identifying,weget
Thelocusof(x1,y1)is(y2−x2)3=4a2x2y2.
Example8.6.23
Anormaltothehyperbola meetstheaxesatQandRandlinesQLand
RLaredrawnatrightanglestotheaxesandmeetatL.ProvethatthelocusofthepointListhehyperbola(a2x2−b2y2)=(a2+b2)2.ProvefurtherthatthelocusofthemiddlepointofQRis4(a2x2−b2y2)=(a2+b2)2.
Solution
LetP(h,k)bethepointonthehyperbola .Theequationofthenormal
at(h,k)is Whenthislinemeetsthex-axisy=0
Therefore,thecoordinatesofQare .ThecoordinatesofRare
.Let(x1,y1)bethecoordinatesofL.Then,
since(h,k)liesonthe
hyperbola.Thelocusof(x1,y1)isa2x2-b2y2=(a2+b2)2.
Let(α,β)bethemidpointofQR.Then,
since(h,k)lieson
thehyperbola.Thelocusof(α,β)isis4(a2x2-b2y2)=(a2+b2)2.
Example8.6.24
Thechordsofthehyperbolax2−y2=a2touchtheparabolay2=4ax.Provethatthelocusoftheirmidpointisthecurvey2(a−y)=x3.
Solution
Let(x1,y1)bethemidpointofthechordofthehyperbola .Itsequation
is
Thelineisatangenttotheparabolay2=4ax.Theconditionis
Thelocusof(x1,y1)isx(x2−y2)=ay2(i.e.)y2(a−y)=x8.
Example8.6.25
Avariabletangenttothehyperbola meetsthetransverseaxisatQand
thetangentatthevertexatR.ShowthatthelocusofthemidpointQRisx(4y2+b2)=ab2.
Solution
Theequationofthetangentat'θ'is
Whenthislinemeetsthetransverseaxis,y=0andx=acosθ.HereQis(acosθ,0).Whenitmeetsthelinex=a,
Let(h,k)bethemidpointofQR.Then,
Hence,thelocusof(h,k)isb2x+4xy2=ab2orx(b2+4y2)=ab2.
Example8.6.26
Showthatthelocusofthemidpointsofthechordsofthehyperbola that
subtendsarightangleatthecentreis
Solution
LetP(x1,y1)bethemidpointofachordofthehyperbols .
Then,theequationofthechordisT=S1
Thechordsubtendsarightangleatthecentreofthehyperbola.Hence,thecombinedequationofthelinesCPandCQis
Since∠QCR=90°,coefficientofx2+coefficientofy2=0.
Example8.6.27
Frompointsonthecirclex2−y2=a2tangentsaredrawntothehyperbolax2−y2
=a2.Provethatthelocusofthemiddlepointsofthechordsofcontactisthecurve(x2−y2)=a2(x2+y2).
Solution
LetP(x1,y1)beapointonthecirclex2+y2=a2.
Let(h,k)bethemidpointofthechordofcontactQRofthetangentsfromPtothehyperbolax2−y2=a2.Thentheequationofchordofcontacttothehyperbolais
xx1−yy1=a2
Theequationofthechordintermsofthemiddlepoint(h,k)is
xh−yk=h2−k
Thesetwoequationsrepresentthesameline.Identifyingthem,weget
Hence,thelocusof(h,k)is(x2−y2)2=a2(x2+y2).
Example8.6.28
Ifthetangentandnormalatanypointofthehyperbola meetonthe
conjugateaxisatQandR,showthatthecircledescribedwithQRasthediameterpassesthroughthefociofthehyperbola.
Solution
Theequationofthetangentandnormalat(x1,y1)onthehyperbola are
ThesetwolinesmeettheconjugateaxisatQandR.Thereforesubstitutex=0in
equations(8.3)and(8.4).ThecoordinatesofQare ThecoordinatesofR
are
TheequationofthecirclewithQRasdiameteris
Substitutingx=±aeandy=0
(i.e.)a2e2−(a2+b2)=0(i.e.)a2e2−a2e2=0whichistrue.Hence,thecirclewithQRasdiameterpassesthroughthefoci.
Exercises
1. Findtheequationofthehyperbolawhosefocusis(1,2),directrix2x+y=1andeccentricity .
Ans.:7x2+12xy−2y2−2x+14y−22=0
2. Showthattheequationofthehyperbolahavingfocus(2,0),eccentricity2anddirectrixx−y=0
isx2+y2−4xy+4=0.
3. Findtheequationofthehyperbolawhosefocusis(2,2),eccentricity anddirectrix3x−4y=1.
Ans.:19x2+44y2−216xy−346x+472y−791=0
4. Findtheequationofthehyperbolawhosefocusis(0,0),eccentricity anddirectrixxcosα+ysin
α=p.
Ans.:16(x2+y2)−25(xcosα+ysinα−p)2=0
5. Findtheequationofthehyperbolawhosecentreis(−3,2)andoneendofthetransverseaxisis
(−3,4)andeccentricityis .
Ans.:4x2−21y2+24x+84y+36=0
6. Findtheequationofthehyperbolawhosefociare(6,4)and(−4,4)andeccentricity2.
Ans.:
7. Findtheequationofthehyperbolawhosecentreis(1,0),onefocusis(6,0)andlengthoftransverseaxisis6.
Ans.:16x2−9y2−32x−128=0
8. Findtheequationofthehyperbolawhosecentreis(3,2),onefocusis(5,2)andonevertexis(4,
2).
Ans.:3x2−y2−18x−4y+20=0
9. Findtheequationofthehyperbolawhosecentreis(6,2),onefocusis(4,2)andeccentricity2.
Ans.:
10. Findthecentre,eccentricityandfociofhyperbola9x2−16y2=144.
Ans.:
11. Findthecentre,fociandeccentricityof12x2−4y2−24x+32y−127=0.Ans.:(1,4),(6,4)and(−4,4)
12. Findthecentre,fociandeccentricityofthehyperbola9x2−4y2−18x+16y−43=0.
Ans.:
13. IfSandS′arethefociofahyperbolaandpisanypointonthehyperbola,showthatS′P−SP=2a.
14. Findthelatusofthehyperbola .
Ans.:
15. Findtheequationofthehyperbolareferredtoitsaxisastheaxisofcoordinateiflengthoftransverseaxisis5andconjugateaxisis4.
Ans.:
16. Findthelatusrectumofthehyperbola4x−9y2=36.
Ans.:
17. Findthecentre,eccentricityandfociofthehyperbolax2−2y2−2x+8y−1=0.
Ans.:
18. Findthecentre,eccentricity,focianddirectrixofthehyperbola16x2−9y2+32x+36y−164=0.
Ans.:
19. Thehyperbola passesthroughtheintersectionofthelines7x+13y−87=0and5x−
8y+7=0anditslatusrectumis Findaandb.
Ans.:
20. Tangentsaredrawntothehyperbola3x2−2y2=6fromthepointPandmakeθ1,θ2withx-axis.Ifthetanθ1tanθ2isaconstant,provethatlocusofPis
2x2−y2=7.
21. Findtheequationoftangentstothehyperbola3x2−4y2=15whichareparalleltoy=2x+k.Findthecoordinatesofthepointofcontact.
Ans.:
22. Tangentsaredrawntothehyperbolax2−y2=c2areinclinedatanangleof45°,showthatthe
locusoftheirintersectionis(x2+y2)2+4a2(x2−y2)=4a4.
23. Provethatthepolarofanypointontheellipse withrespecttothe will
touchtheellipseattheotherendoftheordinatethroughthepoint.
24. Ifthepolarofpoints(x1,y1)and(x2,y2)withrespecttohyperbolaareatrightanglesthenshow
thatb4x1x2+a4y1y2=0.
25. Findthelocusofpolesofnormalchordsofthehyperbola .
26. Chordsofthehyperbola subtendarightangleatoneofthevertices.Showthatthe
locusofpolesofallsuchchordsisthestraightlinex(a2+b2)=a(a2−b2).27. Ifchordsofthehyperbolaareataconstantdistancekfromthecentre,findthelocusoftheirpoles.
Ans.:
28. Obtainthelocusofthepointofintersectionoftangentstothehyperbola which
includesanangleβ.
Ans.:4(a2y2−b2x2+a2b2)=(x2+y2−a2+b2)tan2β
29. Ifavariablechordofthehyperbola isatangenttothecirclex2+y2=c2thenprove
thatthelocusofitsmiddlepointis
30. Showthattheconditionforthelinexcosα+ysinα=βtouchesthehyperbola isa2
cos2α−b2sin2α=p2.31. Provethatthetangentatanypointbisectstheanglebetweenfocaldistancesofthepoint.
32. Provethatthemidpointsofthechordsofthehyperbola paralleltothediametery=
mxbeonthediametera2my=b2x.33. IfthepolarofthepointAwithrespecttoahyperbolapassesthroughanotherpointB,thenshow
thatthepolarBpassesthroughA.
34. Ifthepolarsof(x1,y1)and(x2,y2)withrespecttothehyperbola areatrightangles,
thenprovethat.
35. Provethatthepolarofanypointon withrespecttothehyperbola touches
36. Obtaintheequationofthechordjoiningthepointsθandøonthehyperbolaintheform
.Ifθ−øisaconstantandequalto2α,showthatPQ
touchesthehyperbola
37. Ifacirclewithcentre(3α,3β)andofvariableradiuscutsthehyperbolax2−y2=9a2atthepoints
P,Q,RandSthenprovethatthelocusofthecentroidofthetrianglePQRis(x−2α)2−(y−2β)2=
a2.38. IfthenormalatPmeetsthetransverseaxisinrandtheconjugateaxisingandCFbe
perpendiculartothenormalfromthecentrethenprovethatPF·Pr=CB2andPF·Pg=CF2.39. Showthatthelocusofthepointsofintersectionoftangentsattheextremitiesofnormalchordsof
thehyperbola
40. Findtheequationandlengthofthecommontangentstohyperbolas
Ans.:
41. Tangentsaredrawnfromanypointonhyperbolax2−y2=a2+b2tothehyperbola .
Provethattheymeettheaxesinconjugatepoints.42. Provethatthepartofthetangentatanypointofahyperbolainterceptedbetweenthepointof
contactandthetransverseaxisisaharmonicmeanbetweenthelengthsoftheperpendicularsdrawnfromthefocionthenormalatthesamepoint.
43. Ifthechordjoiningthepointsαandβonthehyperbola isafocalchordthenprove
that wherek≠1.
44. LetthetangentandnormalatapointPonthehyperbolameetthetransverseaxisinTandG
respectively,provethatCT·CG=a2+b2.45. Ifthetangentatthepoint(h,k)tothehyperbolacutstheauxiliarycircleinpointswhoseordinates
arey1andy2thenshowthat
46. IfalineisdrawnparalleltotheconjugateaxisofahyperbolatomeetitandtheconjugatehyperbolainthepointsPandQthenshowthatthetangentsatPandQmeetonthecurve
47. Ifanellipseandahyperbolahavethesameprincipalaxesthenshowthatthepolarofanypointoneithercurvewithrespecttotheothertouchesthefirstcurve.
48. IfthetangentatanypointPonthehyperbola whosecentreisC,meetsthetransverse
andconjugateaxesinT1andT2,thenprovethat(i)CN·CT1=a2and(ii)CM·CT2=−b
2wherePMandPNareperpendicularsinthetransverseandconjugateaxes,respectively.
49. IfPisthelengthoftheperpendicularfromC,thecentreofthehyperbola onthe
tangentatapointPonitandCP=r,provethat
8.7ASYMPTOTES
Definition8.7.1Anasymptoteofahyperbolaisastraightlinethattouchesthehyperbolaatinfinitybutdoesnotliealtogetheratinfinity.
8.7.1EquationsofAsymptotesoftheHyperbola
Lettheequationofthehyperbolabe .Lety=mx+cbeanasymptote
ofthehyperbola.Solvingthesetwoequations,wegettheirpointsofintersection.Thexcoordinatesofthepointsofintersectionaregivenby
Ify=mx+cisanasymptote,thentherootsoftheaboveequationareinfinite.Theconditionsforthesearethecoefficientofx2=0andthecoefficientofx=0,b2−a2m2=0andmca2=0.
Theequationsoftheasymptotesare
Thecombinedequationoftheasymptotesis
Note8.7.1.1:
1. Theasymptotesoftheconjugatehyperbola arealsogivenby Therefore,
thehyperbolaandtheconjugatehyperbolahavethesameasymptotes.
2. Theequationofthehyperbolais
Theequationoftheasymptotesis
Theequationoftheconjugatehyperbolais
3. Theequationoftheasymptotesdiffersfromthatofthehyperbolabyaconstantandtheequationoftheconjugatehyperboladiffersfromthatoftheasymptotesbythesameconstantterm.Thisresultholdsgoodevenwhentheequationsofthehyperbolaanditsasymptotesareinthemostgeneralform.
4. Theasymptotespassthroughthecentre(0,0)ofthehyperbola.
5. Theslopesoftheasymptotesare and
Hence,theyareequallyinclinedtothecoordinateaxes,whicharethetransverseandconjugateaxes.
8.7.2AnglebetweentheAsymptotes
Let2θbetheanglebetweentheasymptotes.Then,
Hence,theanglebetweentheasymptotesis2sec−1(e).
Example8.7.1
Findtheequationoftheasymptotesofthehyperbola3x2−5xy−2y2+17x+y+14=0.
Solution
Thecombinedequationoftheasymptotesshoulddifferfromthatofthehyperbolaonlybyaconstantterm.∴Thecombinedequationoftheasymptotesis
Hence,theasymptotesare3x+y+l=0andx−2y+m=0.
Equatingthecoefficientsofthetermsxandyandtheconstantterms,weget
Solvingthesetwoequations,wegetl=2andm=5.
lm=k
∴k=10.Thecombinedequationoftheasymptotesis(3x+y+2)(x−2y+5)=0.
Example8.7.2
Findtheequationoftheasymptotesofthehyperbolaxy=xh+yk.
Solution
Thecombinedequationoftheasymptotesisxy=xh+yk+norxy−xh−yk−n=0.Theasymptotesarex+l=0andy+m=0.
(x+l)(y+m)=xy−xh−yk−n
Equatingthecoefficientsofthetermsxandyandtheconstantterms,weget
Hence,theequationoftheasymptotesis(x−h)(y−k)=0.
Example8.7.3
Findtheequationtothehyperbolathatpassesthrough(2,3)andhasforitsasymptotesthelines4x+3y−7=0andx−2y=1.
Solution
Thecombinedequationoftheasymptotesis(4x+3y−7)(x−2y−1)=0.Hence,theequationofthehyperbolais(4x+3y−7)(x−2y−1)+k=0.Thispassthrough(2,3).
Hence,theequationofthehyperbolais
Example8.7.4
Findtheequationofthehyperbolathathas3x−4y+7=0and4x+3y+1=0asasymptotesandpassesthroughtheorigin.
Solution
Thecombinedequationoftheasymptotesis
Hence,theequationofthehyperbolais(3x−4y+7)(4x+3y+1)+k=0.Thispassesthroughtheorigin(0,0).∴7+k=0ork=−7Hence,theequationofthehyperbolais
Example8.7.5
Findtheequationsoftheasymptotesandtheconjugatehyperbolagiventhatthe
hyperbolahaseccentricity ,focusattheoriginandthedirectrixalongx+y+
1=0.
Solution
Fromthefocusdirectrixproperty,theequationofthehyperbolais
Thecombinedequationoftheasymptotesis2xy+2x+2y+k=0,wherekisaconstant.Lettheasymptotesbe2x+l=0andy+m=0.Then,
Equatingliketerms,weget2m=2.∴m=1.Similarly,l=2.Aslm=k,wegetk=2.Therefore,theasymphtesofthecombinedequationoftheasymptotesis2xy+
2x+2y+2=0.Theequationoftheasymptotesoftheconjugatehyperbolashoulddifferby
thesameconstant.Theequationoftheasymptotesoftheconjugatehyperbolais2xy+2x+2y+1=0.
Example8.7.6
Derivetheequationsofasymptotes.
Solution
Theequationofthehyperbolais (i.e.)f(x1,y1)=b2x2−a2y2−a2b2=
0.Thisbeingasecond-degreeequation,itcanhavemaximumtwoasymptotes.Asthecoefficientsofthehighestdegreetermsinxandyareconstants,thereisnoasymptoteparalleltotheaxesofcoordinates.Takex=1andy=minthehighestdegreetermsϕ(m)=b2−a2m2.Similarlyϕ(m)=0.Theslopesofthe
obliqueasymptotesaregivenbyϕ2(m)=0.(i.e.)
Also,
Theequationsoftheasymptotesaregivenby
Therefore,thecombinedequationis
Exercises
1. Provethatthetangenttothehyperbolax2−3y2=3at whenassociatedwiththetwo
asymptotesformanequilateraltrianglewhoseareais squareunits.
2. Provethatthepolarofanypointonanyasymptoteofahyperbolawithrespecttothehyperbolaisparalleltotheasymptote.
3. Provethattherectanglecontainedbytheinterceptsmadebyanytangenttoahyperbolaonitsasymptotesisconstant.
4. Fromanypointofthehyperbolatangentsaredrawntoanotherwhichhasthesameasymptotes.Showthatthechordofcontactcutsoffaconstantareafromtheasymptotes.
5. Findtheequationofthehyperbolawhoseasymptotesarex+2y+3=0and3x+4y+5=0andwhichpassesthroughthepoint(1,−1)
Ans.:(x+2y−13)(3x+4y+3)−8=0
6. Findtheasymptotesofthehyperbola3x2−5xy−2y2+5x+11y−8=0.Ans.:x−2y+3=03x+y−4=0
7. Provethatthelocusofthecentreofthecirclecircumscribingthetriangleformedbythe
asymptotesofthehyperbola andavariabletangentis
8. Findtheequationoftheasymptotesofthehyperbola9y2−4x2=36andobtaintheproductofthe
perpendiculardistanceofanypointonthehyperbolafromtheasymptotes.
9. Showthatthelocusofthepointofintersectionoftheasymptoteswiththedirectricesofthe
hyperbola isthecirclex2+y2=a2.
10. LetCbethecentreofahyperbola.ThetangentatPmeetstheaxesinQandRandtheasymptotesinLandM.ThenormalatPmeetstheaxesinAandB.ProvethatLandMlieonthecircleOABandQandRareconjugatewithrespecttothecircle.
11. IfalinethroughthefocusSdrawnparalleltotheasymptotes ofthehyperbola
meetsthehyperbolaandthecorrespondingdirectrixatPandQthenshowthatSQ=2·
SP.
12. Findtheasymptotesofthehyperbola andshowthatthestraightlineparalleltoan
asymptotewillmeetthecurveinonepointatinfinity.13. Provethattheproductoftheinterceptsmadebyanytangenttoahyperbolaonitsasymptotesisa
constant.14. Ifaseriesofhyperbolasisdrawnhavingacommontransverseaxisoflength2athenprovethatthe
locusofapointPoneachhyperbola,suchthatitsdistancefromoneasymptoteisthecurve(x2−
b2)2=4x2(x2−a2).
8.8CONJUGATEDIAMETERS
Locusofmidpointsofparallelchordsofthehyperbolais
Let(x1,y1)bethemidpointofachordofthehyperbola
Thenitsequationis
Theslopeofthischordis
Letthischordbeparalleltoy=mx.
Then
Thelocusof(x1,y1)is ,whichisastraightlinepassingthroughthe
origin.
Ify=m′xbisectsallchordsparalleltoy=mxthen By
symmetry,wenotethaty=mxwillbisectallchordsparalleltoy=m′x.
Definition8.8.1Twodiametersaresaidtobeconjugateifeachbisectschordsparalleltotheother.Theconditionofthediametersy=mxandy=m′xtobe
conjugatediametersis
Note8.8.2Thesediametersarealsoconjugatediametersoftheconjugate
hyperbola since
Property8.8.1
Ifadiametermeetsahyperbolainrealpoints,itwillmeettheconjugatehyperbolainimaginarypointsanditsconjugatediameterwillmeetthehyperbolainimaginarypointsandtheconjugatehyperbolainrealpointsandviceversa.
Proof
Lettheequationofthehyperbolabe
Thentheequationoftheconjugatehyperbolais
Lety=mxandy=m′xbeapairofconjugatediametersofthehyperbola(8.5).Then
Thepointsofintersectionofy=mxandthehyperbola(8.5)aregivenby
Sincethehyperbolameetsy=mxinrealpointsfrom(8.8)b2−a2m2>0.Thepointsofintersectionof(8.6)withy=mxaregivenby
Therefore,y=mxmeetstheconjugatehyperbolainimaginarypoints.Thepointsofintersectionofy=m′xwiththehyperbola(8.5)aregivenby
Theconjugatediametermeetsthehyperbolainimaginarypoints.Alsoitsintersectionwiththeconjugatehyperbolaisgivenby
y=m′xmeetstheconjugatehyperbolainrealpoints.
Property8.8.2
IfapairofconjugatediametersmeetthehyperbolaanditsconjugatehyperbolainPandD,respectivelythenCP2−CD2=a2−b2.
Proof
LetPbethepoint(asecθ,btanθ)ThenDwillhavecoordinates(−atanθ,−bsecθ).ThenCP2=a2sec2θ+b2tan2θCD2=a2tan2θ+b2sec2θCP2−CD2=a2(sec2θ−tan2θ)−b2(sec2θ−tan2θ)=a2−b2
Property8.8.3
Theparallelogramformedbythetangentsattheextremitiesofconjugatediametersofhyperbolahasitsverticeslyingontheasymptotesandisofconstantarea.
Proof
LetPandDbepoints(asecθ,btanθ)and(atanθ,bsecθ)onthehyperbolaanditsconjugate.
ThenD′andP′are(−atanθ,−bsecθ)and(−asecθ,−btanθ),respectively.Theequationsoftheasymptotesare
TheequationsofthetangentsatP,P′,D,D′are
respectively.ClearlythetangentsatPandP′areparallelandalsothetangentsatDandD′areparallel.Solving(8.9)and(8.11)wegetthecoordinatesofDare[a(secθ+tanθ),b[secθ+tanθ)].
Thisliesontheasymptote .
Similarlytheotherpointsofintersectionalsolieontheasymptotes.
TheequationsofPCP′andDCD′are
Lines(8.11),(8.12)and(8.13)areparallelandalsothelines(8.9),(8.10)and(8.11)areparallel.Therefore,areaofparallelogramABCD=4areaofparallelogramCPAD.
Example8.8.1
Ifapairofconjugatediametersmeethyperbolaanditsconjugate,respectivelyinPandDthenprovethatPDisparalleltooneoftheasymptotesandisbisectedbytheotherasymptote.
Solution
Lettheequationofthehyperbolabe
Theequationoftheconjugatehyperbolais
Theasymptotesofthehyperbola(1)are
LetPbethepoint(asecθ,btanθ).ThenDisthepoint(atanθ,bsecθ).
TheslopeofthechordPDis Theslopeofthe
asymptote(8.18)PDisparalleltotheasymptote(8.18).
ThemidpointofPDisa .Thispointliesonthe
asymptotesgivenby(8.17).Therefore,PDisbisectedbytheotherasymptote.
Example8.8.2
Inthehyperbola16x2−9y2=144findtheequationofthediameterconjugatetothediameterx=2y.
Solution
Theequationofthehyperbolais16x2−9y2=144
Theslopeofthelinex=2yis
Ifmandm′aretheslopesoftheconjugatediametersthen
Therefore,theequationoftheconjugatediameteris or32x−9y=0.
Example8.8.3
FindtheconditionthatthepairoflinesAx2+2Hxy+By2=0tobeconjugate
diametersofthehyperbola
Solution
LetthetwostraightlinesrepresentedbyAx2+2Hxy+By2=0bey=m1xandy=m2x.Then
Iftheselinesaretheconjugatediametersofthehyperbolathen
From(8.19)and(8.20)
Property8.8.4
Anytwoconjugatediametersofarectangularhyperbolaareequallyinclinedtotheasymptotes.
Proof
Lettheequationoftherectangularhyperbolabex2−y2=a2.Theequationoftheasymptotesisx2−y2=0.
Lety=mxand beapairofconjugatediametersoftherectangular
hyperbolabex2−y2=a3.Thenthecombinedequationoftheconjugatediametersis
Thecombinedequationofthebisectorsoftheanglesbetweenthesetwolinesis
Thisisthecombinedequationoftheasymptotes.Therefore,theasymptotesbisecttheanglebetweentheconjugatediameter.
8.9RECTANGULARHYPERBOLA
Definition8.9.1Ifinahyperbolathelengthofthesemi-transverseaxisisequaltothelengthofthesemi-conjugateaxis,thenthehyperbolaissaidtobearectanglehyperbola.
8.9.1EquationofRectangularHyperbolawithReferencetoAsymptotesasAxes
Inarectangularhyperbola,theasymptotesareperpendiculartoeachother.Sincetheaxesofcoordinatesarealsoperpendiculartoeachother,wecantaketheasymptotesasthex-andy-axes.Thentheequationsoftheasymptotesarex=0andy=0.Thecombinedequationoftheasymptotesisxy=0.Theequationofthehyperbolawilldifferfromthatofasymptotesonlybya
constant.Hence,theequationoftherectangularhyperbolaisxy=kwherekisaconstanttobedetermined.LetAA′bethetransverseaxisanditslengthbe2a.Then,AC=CA′=a.DrawALperpendiculartox-axis.Sincetheasymptotes
bisecttheanglebetweentheaxes,
ThecoordinatesofAare Sinceitliesontherectangularhyperbolaxy=
k,weget Hence,theequationoftherectangularhyperbolais orxy=
c2where
Note8.9.1.1:Theparametricequationsoftherectangularhyperbolaxy=c2arex
=ctand
8.9.2EquationsofTangentandNormalat(x1,y1)ontheRectangularHyperbolaxy=c2
Theequationofrectangularhyperbolaisxy=c2.Differentiatingwithrespecttox,weget
Theequationofthetangentat(x1,y1)is
since
x1y1=c2.
Theslopeofthenormalat(x1,y1)is
Theequationofthenormalat(x1,y1)is
8.9.3EquationofTangentandNormalat ontheRectangularHyperbola
xy=c2
Theequationoftherectangularhyperbolaisxy=c2.Differentiatingwithrespecttox,weget
slopeofthetangentat
Theequationofthetangentatis is
Theslopeofthenormalat‘t’is−t2.Theequationofthenormalat‘t’is
Dividingbyt,weget
8.9.4EquationoftheChordJoiningthePoints‘t1’and‘t2’ontheRectangularHyperbolaxy=c2andtheEquationoftheTangentatt
Thetwopointsare Theequationsofthechordjoiningthetwo
pointsare
Crossmultiplying,weget
Thischordbecomesthetangentat‘t’ift1=t2=t.Hence,theequationofthetangentat‘t’isx+yt2=2ct.
8.9.5Properties
Anytwoconjugatediametersofarectangularhyperbolaareequallyinclinedtotheasymptotes.
ProofLettheequationoftherectangularhyperbolabex2−y2=a2.The
equationoftheasymptotesisx2−y2=0.Lety=mxand beapairof
conjugatediametersoftherectangularhyperbolax2−y2=a2.Then,thecombinedequationoftheconjugatediametersis
Thecombinedequationofthebisectorsoftheanglesbetweenthesetwolinesis
whichisthecombinedequationoftheasymptotes.
Therefore,theasymptotesbisecttheanglebetweentheconjugatediameter.
8.9.6ResultsConcerningtheRectangularHyperbola
1. Theequationofthetangentat(x1,y1)ontherectangularhyperbolaxy=c2is
2. Theequationofthenormalat(x1,y1)is
3. Theequationofthepairoftangentsfrom(x1,y1)is(xy1+yx1−2c2)2=4(xy−c2)(x1y1−c
2).4. Theequationsofthechordhaving(x1,y1)asitsmidpointisxy1+yx1=2x1y1.
5. Theequationofthechordofcontactfrom(x1,y1)isxy1+yx1=2c2.
8.9.7ConormalPoints—FourNormalfromaPointtoaRectangularHyperbola
Let(x1,y1)beagivenpointandtbethefootofthenormalfrom(x1,y1)ontherectangularhyperbolaxy=c2.
Theequationofthenormalattis
Sincethisnormalpassesthrough(x1,y1),
Thisisafourth-degreeequationintandtherearefourvaluesoft(realorimaginary).Correspondingtoeachvalueoftthereisanormal,andhencetherearefournormalsfromagivenpointtotherectangularhyperbola.
Note8.9.7.1:Ift1,t2,t3andt4arethefourpointsofintersection,then
8.9.8ConcyclicPointsontheRectangularHyperbola
Lettheequationoftherectangularhyperbolabexy=c2.Lettheequationofthecirclebex2+y2+2gx+2fy+k=0.
Let beapointofintersectionofrectangularhyperbolaandthecircle.
Then,thepoint alsoliesonthecircle.Substituting inthe
equationofthecircleweget
Thisisafourthdegreeequationint.Foreachvalueoft,thereisapointofintersection(realorimaginary).Hence,therearefourpointsofintersectionforarectangularhyperbolawiththecircle.
Note8.9.8.1:Ift1,t2,t3andt4arethefourpointsofintersection,then
Example8.9.1
Showthatthelocusofpoleswithrespecttotheparabolay2=4axoftangentstothehyperbolax2−y2=a2istheellipse4x2+y2=4a2.
Solution
Let(x1,y1)bethepolewithrespecttotheparabolay2=4ax.Then,thepolarof
(x1,y1)is
Thisisatangenttotherectangularhypherbolax2−y2=a2.Theconditionfortangencyis
Theconditionfortangencyis
Thelocusof(x1,y1)is4x2+y2=4a2whichisanellipse.
Example8.9.2
Pisapointonthecirclex2+y2=a2andPQandPRaretangentstothehyperbolax2−y2=a2.ProvethatthelocusofthemiddlepointofQRisthecurve(x2−y2)2=a2(x2+y2).
Solution
LetP(x1,y1)beapointonthecirclex2+y2=a2
SincePQandPRaretangentsfromPtotherectangularhyperbolax2–y2=a2,QRisthechordofcontactsoftangentsfromP(x1,y1).Therefore,itsequationisxx1+yy1=a2.Let(h,k)bethemidpointofQR.Itsequationisxh–yk=h2–k2.Thesetwoequationsrepresentthesameline.Therefore,identifyingthem,weget
Since
Thelocusof(h,k)is(x2–y2)2=a2(x2+y2).
Example8.9.3
Provethatthelocusofpolesofallnormalchordsoftherectangularhyperbolaxy=c2isthecurve(x2–y2)+4c2xy=0.
Solution
Let(x1,y1)bethepoleofthenormalchordofrectangularhyperbolaxy=c2.Thepolesof(x1,y1)isxy1+yx1=2c2.Letthechordbenormalatt.The
equationofthenormalattis Thesetwoequationsrepresentthe
samestraightline.Identifyingthem,weget
Also
Thelocusof(x1,y1)is(x2–y2)2+4c2xy=0.
Example8.9.4
IfPisanypointontheparabolax2+16ay=0,provethatthepolesofPwithrespecttorectangularhyperbolaxy=2a2willtouchtheparabolay2=ax.
Solution
Let(x1,y1)beanypoint.ThepolarofPwithrespecttothehyperbolaisxy1+x1y
=4a2(i.e.) Thisisatangenttotheparabolay2=ax.Thecondition
is
Thelocusof(x1,y1)isx2+16ay=0.
Example8.9.5
Atangenttotheparabolax2=4aymeetsthehyperbolaxy=c2atPandQ.ProvethatthemiddlepointofPQliesonafixedparabola.
Solution
Let(x1,y1)bethemidpointofthechordPQoftherectangularhyperbolaxy=c2.TheequationofchordPQis
Thisisatangenttotheparabolax2=4ay.Therefore,theconditionis
(i.e.) Thelocusof(x1,y1)is2x2+ay=0,whichisafixed
parabola.
Example8.9.6
Findthelocusofmidpointsofchordsofconstantlength2loftherectangularhyperbolaxy=c2.
SolutionLetR(x1,y1)bethemidpointofthechordPQ.Lettheequationofthechord
Anypointonthislineisx=x1+rcosθ,y=y1+rsinθ.Ifthispointliesontherectangularhyperbolaxy=c2,weget(x1+rcosθ)(y1+rsinθ)=c2.
Thisisaquadraticequationinr.ThetwovaluesofrarethedistancesRPandRQwhichareequalinmagnitudebutoppositeinsign.Theconditionforthisisthecoefficientofrisequaltozero.
Then,equation(8.21)becomes
Fromequation(8.22),
Substitutingtheseinequation(8.23),weget
butr=l.
Therefore,thelocusof(x1,y1)is(x2+y2)(xy–c2)–l2xy=0.
Example8.9.7
IfPP'isadiameteroftherectangularhyperbolaxy=c2showthatthelocusoftheintersectionoftangentsatPwiththestraightlinethroughP′paralleltoeitherasymptoteisxy+3c2=0.
Solution
LetPbethepoint ThenP′isthepoint
TheequationofthetangentatPisx+yt2=2ct.
TheequationofthestraightlineP'Rparalleltox-axisis
Let(x1,y1)bethepointofintersectionofthesetwolines.Then
Substitutinginequation(8.24),
Thelocusof(x1,y1)isxy+3c2=0.
Example8.9.8
Thetangentstotherectangularhyperbolaxy=c2andtheparabolay2=4axattheirpointofintersectionsareinclinedatanglesαandβ,respectively,tothex-axis.Showthattanα+2tanβ=0.
Solution
Let(x1,y1)bethepointofintersectionoftherectangularhyperbolaxy=c2andtheparabolay2=4ax.Theequationoftangentat(x1,y1)totheparabolaisyy1=2a(x+x1).Theequationoftangenttotherectangularhyperbolaisxy1+yx1=2c2.
Theslopeofthetangenttotheparabolais
Theslopeofthetangenttothetangenttotherectangularhyperbolais
since(x1,y1)liesontheparabolay2=4ax.∴tanα+2tanβ=0
Example8.9.9
Ifthenormaltotherectangularhyperbolaxy=c2atthepointtasitintersecttherectangularhyperbolaatt1thenshowthatt3t1=–1.
Solution
Theequationofthenormalattis
Theequationofthechordjoiningthepointstandt1isx+ytt1=c(t+t1).Thesetwoequationsrepresentthesamestraightline.Identifyingthem,weget
Example8.9.10
Showthattheareaofthetriangleformedbythetwoasymptotesoftherectangularhyperbolaxy=c2andthenormalat(x1,y1)onthehyperbolais
Solution
Theequationofthenormalat(x1,y1)is
Whenthenormalmeetsthex-axis,y=0.
Whenthenormalmeetsy-axis,x=0
Theareaofthetriangle
(i.e.) sincex1y1=c2andignoringthenegativesign.
Example8.9.11
Iffourpointsbetakenonarectangularhyperbolasuchthatthechordjoininganytwoisperpendiculartothechordjoiningtheothertwoandα,β,γ,δaretheinclinationsofthestraightlinesjoiningthesepointstothecentre.provethattanαtanβtanγtanδ=1.
Solution
Lett1,t2,t3,andt4befourpointsP,Q,R,andSontherectangularhyperbolaxy=c2.Theequationofthechordjoiningt1andt2isx+yt1t2=c(t1+t2).
Theslopeofthischordis
Similarly,theslopeofthechordjoiningt3andt4is
Sincethesetwochordsareperpendicular,
TheslopeofthelineCPis
Similarly,
fromequation(8.26)
Example8.9.12
IfthenormalsatthreepartsP,QandRonarectangularhyperbolaintersectatapointSonthecurvethenprovethatthecentreofthehyperbolaisthecentroidofthetrianglePQR.
Solution
Ifthenormalattmeetsthecurveatt'thent2t'=–1.
Thisisacubicequationint.Ift1,t2andt3aretherootsofthisequationtheycanberegardedastheparametersofthepointsP,QandR,thenormalsatthesepointsmeetatt'whichisS.Fromequation(8.27),wegett1+t2+t3=0andt1t2+t2t3+t3t1=0.Let(h,k)bethecentroidofΔPQR.
Then
Thecentroidisthecentreoftherectangularhyperbola.
Example8.9.13
Showthatfournormalscanbedrawnfromapoint(h,k)totherectangularhyperbolaxy=c2andthatitsfeetformatriangleanditsorthocentre.
Solution
Theequationofthenormalattis
(i.e.)ct4–xt3+yt–c=0
Sincethispassesthrough(h,k),ct4–ht3+kt–c=0.Thisisafourthdegreeequationint.Itsrootsaret1,t2,t3andt4whicharethe
feetofthefournormalsfrom(h,k).
Ift1,t2,t3andt4arethepointsP,Q,RandSontherectangularhyperbolaxy=
c2,itcanbeshownthattheorthocentreofthetriangleis
Thispointis ist1t2t3t4=–1.
∴ThefourpointsP,Q,RandSformatriangleanditsorthocentre.
Example8.9.14
Provethatfromanypoint(h,k)fournormalscanbedrawntotherectangularhyperbolaxy=c2andthatifthecoordinatesofthefourfeetofthenormalsP,Q,RandSbe(xr,yr),r=1,2,3,4.Then(i)x1+x2+x3+x4=h,y1+y2+y3+y4=kand(ii)x1x2x3x4=y1y2y3y4=–c4.
Solution
Theequationofthenormalattis
Sincethispassesthrough(h,k)
Theformvaluesoftcorrespondtothefeetofthefournormalsfromthepoint(h,k).Ift1,t2,t3andt4arethefourfeetofthenormalsthentheyaretherootsoftheaboveequation.
Fromequation(8.28),c(t1+t2+t3+t4)=h
(i.e.)x1+x2+x3+x4=h
Dividingequation(8.30)byequation(8.31),weget
Example8.9.15
Provethatthefeetoftheconcurrentnormalsontherectangularhyperbolaxy=c2whichmeetsat(h,k)lieonanotherrectangularhyperbolawhichpasses
through(0,0)and(h,k).
Solution
Theequationofthenormalat(x1,y1)is
Sincethispassesthrough(h,k),
Thelocusof(x1,y1)isx2–y2–hx+ky=0.Clearlythisisarectangularhyperbolapassingthrough(0,0)and(h,k).
Example8.9.16
IfarectangularhyperbolawhosecentreisciscutbyanycircleofradiusrinfourpointsP,Q,R,SthenprovethatCP2+CQ2+CR2+CS2=4r2.
Solution
Lettheequationoftherectangularhyperbolabe
Lettheequationofthecirclebe
Solvingthesetwoequations,wegettheirpointsofintersections
Substitutinginequation(8.33),
Ifx1,x2,x3,x4aretheabscissaeofthefourpointsofintersectionx1+x2+x3+x4=–2g.
Example8.9.17
A,B,CandDarefourpointsofintersectionofacircleandarectangularhyperbola.IfABpassesthroughthecentreofthehyperbola,showthatCDpassesthroughthecentreofthecircle.
Solution
Lettheequationoftherectangularhyperbolabexy=c2.Lettheequationofthecirclebex2+y2+2gx+2fy+k=0.LetA,B,CandDbethepointst1,t2,t3andt4,respectively.Whenthecircleandrectangularhyperbolaintersectweknowthat
TheequationofthechordABisx+yt1t2=c(t1+t2).SinceABpassesthrough(0,0),weget
t1+t2=0(8.38)
∴Fromequation(8.34),
usingequation(8.38)
[Fromequations(8.39)and(8.40)]
TheequationofthechordCDisx+yt3t4=c(t3+t4).
Thisstraightlinepassesthroughthepoint(–g,–f).Therefore,CDpassesthroughthecentreofthecircle.
Example8.9.18
ShowthatthroughanygivenpointPintheplaneofxy=c2,fournormalscanbedrawntoit.IfP1,P2,P3andP4arefeetofthesenormalsandCiscentrethen
showthat
Solution
Theequationofthenormalattis
Letpbethepoint(h,k).Sincethenormalpassesthrough(h,k),
Thisbringsafourthdegreeequation,therearefournormalsfromP.
Ift1,t2,t3andt4arethefeetofthenormalsthen
Example8.9.19
TheslopesofthesidesoftriangleABCinscribedinarectangularhyperbolaxy=c2aretanα,tanβandtanγ.IfthenormalsatA,BandCareconcurrentshowthatcot2α+cot2β+cot2γ=0.
Solution
LetA,B,Cbethepointst1,t2andt3respectively.
TheslopeofABis
Example8.9.20
Showthataninfinitenumberoftrianglescanbeinscribedinarectangularhyperbolaxy=c2whosesidestouchtheparabolay2=4ax.
Solution
LetABCbeatriangleinscribedintherectangularhyperbolaxy=c2.LetA,BandCbethepointst1,t2,andt3,respectively.SupposethesidesABandACtouchtheparabolay2=4ax.TheequationofthechordABisx+yt1t2=c(t1+t2).
Thistouchestheparabolay2=4ax.(i.e.)
(i.e.)c(t1+t2)+a(t1t2)2=0
(i.e.) SinceACalsotouchestheparabola,
Fromtheseequations,wenotethatt2,t3aretherootsoftheequation
TheequationofthechordBCisx+yt2t3=c(t2+t3).
ThisequationshowsthatBCtouchestheparabolay2=4ax.SinceABCisanarbitrarytriangleinscribedintherectangularhyperbolaxy=c2thereareinfinitenumberofsuchtrianglestouchingtheparabolay2=4ax.
Exercises
1. Provethattheportionofthetangentinterceptedbetweenbyitsasymptotesisbisectedatthepointofcontactandformatriangleofcontactarea.
2. Ifthetangentandnormaltoarectangularhyperbolamakeinterceptsa1anda2ononeasymptoteandb1andb2ontheotherthenshowthata1a2+b1b2=0.
3. PandQarevariablepointsontherectangularhyperbolaxy=c2suchthatthetangentatQpassesthroughthefootoftheordinateofP.ShowthatthelocusoftheintersectionofthetangentsatPandQisahyperbolawiththesameasymptotesasthegivenhyperbola.
4. Ifthelinesx–α=0andy−β=0areconjugatelineswithrespecttothehyperbolaxy=c2then
provethatthepoint(α,β)isonthehyperbolaxy–2c2=0.
5. Ifthechordsofthehyperbolax2−y2=a2touchtheparabolay2=4axthenprovethatthelocusof
theirmiddlepointsisthecurvey2(x–a)=x3.
6. IfPQandPRaretwoperpendicularchordsoftherectangularhyperbolaxy=c2thenshowthatQRisparalleltothenormalatP.
7. Ifthepolarofapointwithrespecttotheparabolay2=4axtouchestheparabolax2=4by,showthatthepointshouldlieonarectangularhyperbola.
8. Showthatthenormalattherectangularhyperbolaxy=c2atthepoint meetsthecurve
againatthepoint .ShowthatPQvariesasCP2whereCisthecentre.
9. IfPQisachordoftherectangularhyperbolaxy=c2whichisthenormalatPshowthat3CP2+
CQ2=PQ2whereCisthecentreoftheconic.10. Tworectangularhyperbolasaresuchthattheaxesofonearealongtheasymptotesoftheother.
Findthedistancebetweenthepointofcontactofacommontangenttothem.11. Provethatanylineparalleltoeitheroftheasymptotesofahyperbolashouldmeetitinonepointat
infinity.12. ThetangentatanypointofthehyperbolameetstheasymptotesatQandR.ShowthatCQ·CRisa
constant.13. Provethatthelocusofthecentreofthecirclecircumscribingthetriangleformedbythe
asymptotesofthehyperbola andavariabletangentis4(a2x2–b2y2)=a2+b2.
14. Showthatthecoordinatesofthepointofintersectionoftwotangentstoarectangularhyperbolaareharmonicmeansbetweenthecoordinatesofthepointofcontact.
15. IfthenormalsatA,B,CandDtotherectangularhyperbolaxy=c2meetinP(h,k)thenprovethat
PA2+PB2+PC2+PD2=3(h2+k2).
16. If(ctanϕ,ccotϕ)beapointontherectangularhyperbolaxy=c2thenshowthatthechordsthroughthepointsϕandϕ'whereϕ+ϕ'isaconstantpassesthroughafixedpointontheconjugateaxisofthehyperbola.
17. Provethatthepoleswithrespecttothecirclex2+y2=a2ofanytangenttotherectangular
hyperbolaxy=c2liesonrectangularhyperbola4c2xy=c2.18. Ifanormaltoarectangularhyperbolamakesanacuteangleθwithitstransverseaxisthenprove
thattheacuteangleatwhichitcutsthecurveagainiscot–1(2tan2θ).
19. Ifacirclecutstherectangularhyperbolaxy=c2infourpointsthenprovethattheproductofthe
abscissaeofthepointsisc4.
20. Lettherectangularhyperbolaxy=c2iscutbyacirclepassingthroughitscentreCinfourpoints
P,Q,RandS.Ifp,qbetheperpendicularsfromconPQ,RSthenshowthatpq=c2.
21. Ifatriangleisinscribedinarectangularhyperbolaxy=c2andtwoofitssidesareparalleltoy=
m1xandy=m2xthenprovethatthethirdsidetouchesthehyperbola4m1m2xy=c2(m1+m2)
2.
22. Ifacirclecutstherectangularhyperbolaxy=c2inP,Q,RandSandtheparametersofthesefourpointsbet1,t2,t3andt4,respectivelythenprovethatthecentreofthemeanpositionofthese
pointsbisectthedistancebetweenthecentresofthetwocurves.
23. Ifthreetangentsaredrawntotherectangularhyperbolaxy=c2atthepoints(xi,yi),i=1,2,3andformatrianglewhosecircumcirclepassesthroughthecentreofthehyperbolathenshowthat
andthatthecentreofthecircleliesonthehyperbola.
24. Ifacirclewithfixedcentre(3p,3q)andofvariableradiuscutstherectangularhyperbolax2–y2=
9c2atthepointsP,Q,RandSthenshowthatthelocusofthecentroidofthetrianglePQRisgiven
by(x–2p)2–(y–2q)2=a2.25. Showthatthesumoftheeccentricanglesofthefourpointsofintersectionofanellipseanda
rectangularhyperbolawhoseasymptotesareparalleltotheaxesoftheellipseisanoddmultipleofπ.
26. Iffromanypointonthelinelx+my+1=0tangentsPQ,PRaredrawntotherectangular
hyperbola2xy=c2andthecirclePQRcutsthehyperbolaagaininTandT'thenprovethatTT'
touchestheparabola(l2+m2)(x2+y2)=(lx+my+1)2.27. Ifacirclecutstwofixedperpendicularlinessothateachinterceptisofgivenlengththenprove
thatthelocusofthecentreofthecircleisarectangularhyperbola.28. IfAandBarepointsontheoppositebranchesofarectangularhyperbola.ThecircleonABas
diametercutsthehyperbolaagainatCandDthenprovethatCDisadiameterofthehyperbola.
29. IfA,BandCarethreepointsontherectangularhyperbolaxy=d2whoseabscissaearea,bandc
respectivelythenprovethattheareais andtheareaofthetriangle
enclosedbytangentsatthesepointsis
30. Iffourpointsonarectangularhyperbolaxy=c2lieonacircle,thenprovethattheproductoftheir
abscissaeisc4.31. Ifx1,x2,x3andx4betheabscissaeoftheangularpointsandtheorthocentreofatriangle
inscribedinxy=c2thenprovethatx1x2x3x4=–c4.
32. Showthatthelengthofthechordoftherectangularhyperbolaxy=c2whichisbisectedatthe
point(h,k)is
33. ProvethatthepointofintersectionoftheasymptotesofarectangularhyperbolawiththetangentatanypointPandoftheaxeswiththenormalatPareequidistantfromP.
34. IfPisanypointonarectangularhyperbolawhoseverticesareAandA'thenprovethatthebisectorsofangleAPA'areparalleltotheasymptotesofthecurve.
35. LetQCQ'isadiameterofarectangularhyperbolaandPisanypointonthecurve.ProvethatPQ,
PQ'areequallyinclinedtotheasymptotesofthehyperbola.36. ThroughthepointP(0,b)alineisdrawncuttingthesamebranchoftherectangularhyperbolaxy=
c2inQandRsuchthatPQ=QR.Showthatitsequationis9c2y+2b2x=9bc2.
37. Ifarectangularhyperbolaxy=c2iscutbyacirclepassingthroughitscentreOinpointsA,B,CandDwhoseparametersaret1,t2,t3andt4thenshowthat(t1+t2)(t3+t4)+t1t2+t3t4=0and
deducethattheproductoftheperpendicularfromOonABandCDisc2.
Chapter9
PolarCoordinates
9.1INTRODUCTION
Acoordinatesystemrepresentsapointinaplanebyanorderedpairofnumberscalledcoordinates.EarlierweusedCartesiancoordinateswhicharedirecteddistancesfromtwoperpendicularaxes.NowwedescribeanothercoordinatesystemintroducedbyNewtoncalledpolarcoordinateswhichismoreconvenientforsomespecialpurposes.
9.2DEFINITIONOFPOLARCOORDINATES
Wechooseapointintheplaneanditiscalledthepole(ororigin)andisdenotedbyO.Thenwedrawaray(halfline)startingatOcalledpolaraxis.Thisisusuallydrawnhorizontallytotherightandcorrespondstopositivex-axisinCartesiancoordinates.
LetPbeanypointintheplaneandrbethedistancefromOtoP.Letθbetheangle(usuallymeasuredinradians)betweenthepolaraxisandthelineOP.ThenthepointPisrepresentedbytheorderedpair(r,θ)and(r,θ)arecalledthepolarcoordinatesofthepointP.Weusetheconventionthatanangleispositiveifmeasuredintheanti-clockwisedirectionfromthepolaraxisandnegativeintheclockwisedirection.
IfPcoincideswithOthenr=θ.Then(r,θ)representthecoordinatesofthepoleforanyvalueofθ.Letusnowextendthemeaningofpolarcoordinates(r,θ)whenrisnegative,agreeingthatthepoints(−r,θ)and(r,θ)lieonthesamelinethroughOandatthesamedistance|r|fromObutonoppositesidesofO.Ifr>0,thepoint(r,θ)liesonthesamequadrantasθ.Ifr<0,thenitliesinthequadrantoftheoppositesideofthepole.Wenotethatthepoint(r,θ)representsthesamepointas(r,θ+π)
Example9.2.1
Representthefollowingpolarcoordinatesinthepolarplane:
Solution
Thecoordinates, and arerepresentedbypointsin
thefollowingdiagram:
InCartesiansystemofcoordinates,everypointhasonlyonerepresentation.Butinpolarcoordinatessystemeachpointhasmanyrepresentations,for
example,point isalsorepresentedby ,etc.
Ingeneral,thepoint(r,θ)isalsorepresentedby(r,θ+2nπ)or(−r,θ+2n+1π)wherenisanyinteger.
9.3RELATIONBETWEENCARTESIANCOORDINATESANDPOLARCOORDINATES
If(x,y)istheCartesiancoordinatesand(r,θ)arethepolarcoordinatesofthe
pointP,then and
Therefore,thetransformationsfromonesystemtoanotheraregivenbyx=rcosθ,y=rsinθ.Tofindrfromxandy,weusetherelationr2=x2+y2andθisgivenby
Wehavealreadystudiedthedistancebetweentwopoints,areaofatriangle,equationsofastraightline,equationstoacircleandequationofconicsinCartesiancoordinatessystem.Letusnowderivetheresultsinpolarcoordinatesystem.
9.4POLAREQUATIONOFASTRAIGHTLINE
ThegeneralequationofastraightlineinCartesiancoordinatesisAx+By+C=0,whereA,BandCareconstants.Let(r,θ)bepolarcoordinatesofapointandthex-axisbetheinitialline.Thenforanypoint(x,y)onthestraightlinex=rcosθ,y=rsinθ.Substitutingtheseintheequationofstraightline,weget
Thiscanbewrittenintheform
whereA,Bandlareconstants.Therefore,equation(9.1)isthegeneralequationofastraightlineinpolarcoordinates.
9.5POLAREQUATIONOFASTRAIGHTLINEINNORMALFORM
Lettheoriginbethepoleandthex-axisbetheinitialline.DrawONperpendiculartothestraightline.LetON=pand∠XON=α.
Thisisthepolarequationoftherequiredstraightline.
Note9.5.1:Polarequationofthestraightlineperpendicularto
isoftheform
or ,wherekisaconstant.
Note9.5.2:Thepolarequationofthestraightlineparallelto
is ,wherekisaconstant.
Note9.5.3:Theconditionforthestraightlines and
tobeperpendiculartoeachotherisAA1+BB1=0.Thisresult
canbeeasilyseenfromtheircartesianequations.
Note9.5.4:Ifthelineisperpendiculartotheinitiallinethenα=0orπ.Therefore,theequationofthestraightlineisrcosθ=porrcosθ=−p.
Note9.5.5:Ifthelineisparalleltotheinitiallinethen or .Inthiscase
theequationofthelineis
Example9.5.6
FindtheequationofthestraightlinejoiningthetwopointsP(r1,θ1)andQ(r2,θ2).
Solution
LetR(r,θ)beanypointonthelinejoiningthepointsPandQ.TheareaofthetriangleformedbythepointsP(r1,θ1),Q(r2,θ2)and(r3,θ3)is
Takingr3=randθ3=θ,weget
SincethepointsP,QandRarecollinear,Δ=0.
Dividingbyrr1r2,weget
Thisistheequationoftherequiredstraightline.
Example9.5.7
Findtheslopeofthestraightline
Solution
Theequationofthestraightlineis
Therefore,theslopeofthestraightlineis
Example9.5.8
Findthepointofintersectionofthestraightlines and
.
Solution
Theequationsofthestraightlinesare
Solvingequations(9.2)and(9.3),weget
Therefore,theonlypossibilityis .
Thenfromtheequationofthefirststraightline,weget
Hence,thepointofintersectionofthetwogivenlinesis
Example9.5.9
Findtheequationofthelinejoiningthepoints and anddeducethat
thislinealsopassesthroughthepoint .
Solution
Theequationofthelinejoiningthepoints(r1,θ1)and(r2,θ2)is
Therefore,theequationofthelinejoiningthepoints and is
Hence,thepoint liesonthestraightline.
Example9.5.10
Showthatthestraightlinesr(cosθ+sinθ)=±1andr(cosθ−sinθ)=±1encloseasquareandcalculatethelengthofthesidesofthissquare.
Solution
ConvertingintoCartesianformthefourlinesare
Thesefourlinesformaparallelogramandinx+y=±1,x−y=±1theadjacentlinesareperpendicularandhenceABCDisarectangle.
AlsothedistancebetweenABandCD= .
ThedistancebetweenADandBC= .
Therefore,thesefourlinesformasquare.
Example9.5.11
Findtheanglebetweenthelines and
Solution
Exercises
1. Findtheanglebetweenthelines
i. rcosθ=p,rsinθ=p1
ii.
Ans.:
2. Showthatthepoints and arecollinear.
3. Showthattheequationofanylineparallelto throughthepoleis
4. Findtheequationofthelineperpendicularto andpassingthroughthepoint
(r1,θ1).
Ans.:
9.6CIRCLE
9.6.1PolarEquationofaCircle
LetObethepoleandOXbetheinitialline.LetC(c,α)bethepolarcoordinatesofthecentreofthecircle.LetP(r,θ)beanypointonthecircle.Then∠COP=θ−α.Letabetheradiusofthecircle.
Thisisthepolarequationoftherequiredcircle.
Note9.6.1.1:Ifthepoleliesonthecircumferenceofthecirclethenc=a.Thentheequationofthecirclebecomes,
Note9.6.1.2:Theequationofthecircler=2acos(θ−α)canbewrittenintheformr=Acosθ+BsinθwhereAandBareconstants.
Note9.6.1.3:Ifthepoleliesonthecircumferenceofthecircleandtheinitiallinepassesthroughthecentreofthecirclethentheequationofthecirclebecomes,r=2acosθsinceα=0.
Note9.6.1.4:Supposetheinitiallineisatangenttothecircle.Thenc=acosecα.Therefore,fromequation(9.4)theequationofthecirclebecomes,a2=a2
cosec2α+r2–2arcosecαcos(θ–α)
(i.e.)r2–2racosecαcos(θ–α)+a2cot2α=0
Note9.6.1.5:Supposetheinitiallineisatangentandthepoleisatthepointofcontact.Inthiscaseα=90°.Theequationofthecirclebecomes,r2−2rasinθ=0(or)r=2asinθ.
9.6.2EquationoftheChordoftheCircler=2acosθontheLineJoiningthePoints(r1,θ1)and(r2,θ2).
LetPQbethechordofthecircler=2acosθ.
LetPandQbethepoints(r1,θ1)and(r2,θ2).SincethepointsPandQlieonthecircle
LettheequationofthelinePQbe
SincethepointsPandQlieonthisline
Fromequations(9.6)and(9.7),weget
Hence,fromequation(9.6),wegetp=2acosθ1cosθ2.Hence,fromequation(9.5)theequationofthechordis2acosθ1cosθ2=
.
Note9.6.2.1:Thischordbecomesthetangentatαifθ1=θ2=α.Therefore,theequationofthetangentatαis2acos2α=rcos(θ–2α).
9.6.3EquationoftheNormalatαontheCircler=2αcosθ
SinceONisperpendiculartoPN,
Theequationofthenormalisp=rcos(θ–α).
9.6.4EquationoftheCircleontheLineJoiningthePoints(a,α)and(b,β)astheendsofaDiameter
Since∠APB=90°
Example9.6.1
Showthatthelocusofthefootoftheperpendiculardrawnfromthepoletothetangenttothecircler=2acosθisr=a(l+cosθ).
Solution
LetPbethepoint(r,α).DrawONperpendiculartothetangentatP.
TheequationofthetangentatPis
rcos(θ−2α)=2acos2α
SinceONistheperpendiculardistancefromOonthelinePN,fromthenormalformofthestraightline,weget
ON=p=2acos2αLetthecoordinatesofNbe(r1,θ1),then
Example9.6.2
Showthatthefeetoftheperpendicularsfromtheoriginonthesidesofthetriangleformedbythepointswithvectorialanglesα,β,γandwhichlieonthecircler=2acosθlieonthestraightline2acosαcosβcosγ=rcos(π–α–β–γ).
Solution
Theequationofthecircleisr=2acosθ.LetthevectorialanglesofP,Q,Rbeα,β,γrespectively.TheequationsofthechordPQ,QRandRPare
LetL,MandNbethefeetoftheperpendicularsfromOonthelinesPQ,QRandRPThenfromtheaboveequations,weinferthatthecoordinatesofL,MandN
are
Thesethreepointssatisfytheequation
2acosαcosβcosγ=rcos(θ−α−β−γ)HenceL,MandNliesontheaboveline.
Example9.6.3
Showthatthestraightline touchesthecircler=2acosθifa2
B2+2alA=l2.
Solution
Theequationofthecircleis
Theequationofthestraightlineis
Solvingthesetwoequationswegettheirpointofintersection.
Dividingbycosθ,weget
Iftheline(9.9)isatangentto(9.8)thenthetwovaluesoftanθoftheequation(9.10)areequal.Theconditionforthatisthediscriminantisequaltozero.
Exercises
1. Showthatr=Acosθ+Bsinθrepresentsacircleandfindthepolarcoordinatesofthecentre.
2. Showthattheequationofthecircleofradiusawhichtouchesthelinesθ=0, isr2–2ar(cos
θ+sinθ)+a2=0.Showthatlocusoftheequationr2−2racos2θsecθ−2a2=0consistsofastraightlineandacircle.
3. Findthepolarequationsofcirclespassingthroughthepointswhosepolarcoordinatesare
andtouchingthestraightlineθ=0.
Ans.:r2−r[(a+b)Sinθ±2bcosθ]+c2=0
4. Acirclepassesthroughthepoint(r,θ)andtouchestheinitiallineatadistancecfromthepole.
Showthatitspolarequationis
5. Showthatr2−krcos(θ−α)+kd=0representsasystemofgeneralcirclesfordifferentvaluesofk.Findthecoordinatesofthelimitingpointsandtheequationofthecommonradicalaxis.
6. Findtheequationofthecirclewholecentreis andradiusis2.
Ans.:
7. Findthecentreandradiusofthecircler2–10rcosθ+9=0.Ans.:(5,0);4
8. Provethattheequationtothecircledescribedonthelinejoiningthepoints and as
diameteris
9. Findtheconditionthattheline maybea
i. tangentii. anormaltothecircler=2cosθ.
10. Findtheequationofthecirclewhichtouchestheinitialline,thevectorialangleofthecentrebeingαandtheradiusofthecirclea.
11. Acirclepassesthroughthepoint(r1,θ1)andtouchestheinitiallineatadistancecfromthepole.Showthatitspolarequationis
9.7POLAREQUATIONOFACONIC
Earlierwedefinedparabola,ellipseandhyperbolaintermsoffocusdirectrix.Nowletusshowthatitispossibletogiveamoreunifiedtreatmentofallthesethreetypesofconicusingpolarcoordinates.Furthermore,ifweplacethefocusattheoriginthenaconicsectionhassimplepolarequation.LetSbeafixedpoint(calledthefocus)andXM,afixedstraightline(called
thedirectrix)inaplane.Letebeafixedpositivenumber(calledthe
eccentricity).ThenthesetofallpointsPintheplanesuchthat iscalleda
conicsection.Theconicis
i. anellipseife<1.ii. aparabolaife=1.iii. ahyperbolaife>1.
9.7.1PolarEquationofaConic
LetSbefocusandXMbethedirectrix.DrawSXperpendiculartothedirectrix.LetSbethepoleandSXbetheinitialline.LetP(r,θ)beanypointontheconic;thenSP=r,∠XSP=θ.DrawPMperpendiculartothedirectrixandPNperpendiculartotheinitialline.
LetLSL′bethedoubleordinatethroughthefocus(latusrectum).Thefocusdirectrixpropertyis
Thisistherequiredpolarequationoftheconic.
Note9.7.1.1:Iftheaxisoftheconicisinclinedatanangleαtotheinitialline
thenthepolarequationofconicis
Totracetheconic,
cosθisaperiodicfunctionofperiod2π.Therefore,totracetheconicitisenoughifweconsiderthevariationofθfrom
–πtoπ.Sincecos(–θ)=cosθthecurveissymmetricalabouttheinitialline.Henceitisenoughifwestudythevariationofθfrom0toπ.Letusdiscussthevariouscasesfordifferentvaluesofθ.Case1:Lete=0.Inthiscase,theconicbecomesr=lwhichisacircleofradiuslwithitscentreatthepole.
Case2:Lete=1.Inthiscase,theequationoftheconicbecomes, Whenθ
variesfrom0toπ,1+cosθvariesfrom2to0.
and variesfrom to∞
Theconicinthiscaseisaparabolaandisshownbelow.
Case3:Lete<1.
Asθvariesfrom0toπ,1+ecosθdecreasesfrom1+eto1–e.rincreasesfrom to
Thecurveisclearlyclosedandissymmetricalabouttheinitialline.Theconicisanellipse.
Case4:Lete>1.
Asθvariesfrom0to ,1+ecosθdecreasesfrom(1+e)to1andhencerincreasesfrom tol.
Asθvariesfrom toπ,1+ecosθdecreasesfrom1to(1–e).
Therefore,thereexistsanangleαsuchthat <α<πatwhich1+ecosθ>0.(i.e.)
Hence,asθvariesfrom toα,rincreasesfrom1to∞.Asθvariesfromαtoπ,1+ecosθremains
negativeandvariesfrom0to(1−e).
rvariesfromto−∞to .
Theconicisshownaboveandisahyperbola.
9.7.2EquationtotheDirectrixCorrespondingtothePole
LetQbeanypointonthedirectrix.Letitscoordinatesbe(r,θ).ThenSX=rcosθ
or Sincethisistrueforallpoints(r,θ)onthedirectrix,the
polarequationofthedirectrixis
Note9.7.2.1:Theequationofthedirectrixoftheconic is
.
Thepolarequationoftheconicfordifferentformofdirectrixisgivenbelow.
Note9.7.2.2:Theaboveconicisanellipseife<1,parabolaife=1andhyperbolaife>1.
9.7.3EquationtotheDirectrixCorrespondingtoFocusOtherthanthePole
Let(r,θ)bethecoordinatesofapointonthedirectrixX′M′.
Then
But
Thisistherequiredequationoftheotherdirectrix.
9.7.4EquationofChordJoiningthePointswhoseVectorialAnglesareα−β
andα+βontheConic
Lettheequationoftheconicbe .
LettheequationofthechordPQbe .
Thischordpassesthroughthepoint(SP,α−β)and(SQ,α+β).
Alsothesetwopointslieontheconic ,
Fromequations(9.11)and(9.13),weget
Fromequation(9.12)and(9.14),weget
Subtracting,weget
Fromequation(9.15),weget
TheequationofthechordPQis
9.7.5TangentatthePointwhoseVectorialAngleisαontheConic
Theequationofthechordjoiningthepointswithvectorialanglesα−βandα+
βis .
Thischordbecomesthetangentatαifβ=0.
Theequationoftangentatαis .
9.7.6EquationofNormalatthePointwhoseVectorialAngleisαontheConic
Theequationoftheconicis
Theequationoftangentatαontheconic is
Theequationofthelineperpendiculartothistangentis
.
IfthisperpendicularlineisnormalatP,thenitpassesthroughthepoint(SP,α).
Sincethepoint(SP,α)alsoliesontheconic wehave
Fromequation(9.17),weget .
Hence,theequationofthenormalatαis
9.7.7AsymptotesoftheConicis
Theequationoftheconicis
Theequationofthetangentatαis
Thistangentbecomesanasymptoteifthepointofcontactisatinfinity,thatis,thepolarcoordinatesofthepointofcontactare(∞,α).Sincethispointhastosatisfytheequationoftheconic(9.18)wehavefromequation(9.18),
Theequation(9.19)canbewrittenas
Substituting and wegettheequationofthe
asymptotesas
9.7.8EquationofChordofContactofTangentsfrom(r1,θ1)totheConic
LetQRbethechordofcontactoftangentsfromP(r1,θ1).LetvectorialanglesofQandRbeα−βandα+β.TheequationofthechordQRis
TheequationsoftangentsatQandRare
Thesetwotangentsintersectat(r1,θ1).
Fromtheabovetwoequations,weget
Substitutingthisinequation(9.24),weget
Substitutingequation(9.26)in(9.21),weget
Thisistheequationofthechordofcontact.
9.7.9EquationofthePolarofanyPoint(r1,θ1)withRespecttotheconic
ThepolarofapointwithrespecttoaconicisdefinedasthelocusofthepointofintersectionoftangentsattheextremitiesofavariablechordpassingthroughthepointP(r1,θ1).
LetthetangentsatQandRintersectT.SinceQRisthechordofcontactoftangentsfromT(R,ϕ),itsequationis
SincethispassesthroughthepointP(r1,θ1)wehave
NowthelocusofthepointT(R,ϕ)ispolarofthe(r1,θ1).
Thepolarof(r1,θ1)fromequation(9.28)is
Example9.7.1
Findtheconditionthatthestraightline maybeatangenttothe
conic
Solution
Lettheline touchestheconicatthepoint(r,α).
Thentheequationoftangentat(r,α)is
However,theequationoftangentisgivenas
Equations(9.30)and(9.31)representthesameline.
Identifyingequations(9.30)and(9.31),weget
Squaringandadding,weget(A−e)2+B2=1Thisistherequiredcondition.
Example9.7.2
Showthatinaconic,semilatusrectumistheharmonicmeanbetweenthesegmentsofafocalchord.
Solution
LetPQbeafocalchordoftheconic LetPandQhavethepolar
coordinates(SP,α)and(SQ,α+π).
SincePandQlieontheconic .
Wehave
Addingequations(9.32)and(9.33)
SP,l,SQareinHP(i.e.)listheHMbetweenSPandSQ.
Example9.7.3
Showthatinanyconicthesumofthereciprocalsoftwoperpendicularfocalchordsisaconstant.
Solution
LetPP′andQQ′beperpendicularfocalchordsoftheconic
LetPbethepoint(SP,α).ThevectorialanglesofQ,P′,Q′are
.
SincethepointsP,P′,Q,Q′lieontheconic,
wehave
Example9.7.4
IfachordPQofaconicwhoseeccentricityeandthesemilatusrectuml
subtendsarightangleatthefocusSPthenprovethat
Solution
Lettheequationoftheconicbe .LetthevectorialangleofPbeα.
ThevectorialangleofQis .
SincePandQlieontheconic,
Similarly,
Squaringandadding,weget
Example9.7.5
LetPSQandPS′RbetwochordsofanellipsethroughthefociSandS′.Show
that isaconstant.
Solution
LetthevectorialangleofPbeα.ThenthevectorialangleofQisα+π.SinceP
andQlieontheconic
Similarly,consideringtheotherfocalchordPS′R
Multiplyequation(9.37)by ,weget
Similarlyfromequation(9.38),weget
Addingequations(9.39)and(9.40),weget
Example9.7.6
Provethattheperpendicularfocalchordsofarectangularhyperbolaareequal.
Solution
LetPSP′andQSQ′andbetwoperpendicularfocalchordsofarectangularhyperbola.ThenthevectorialanglesofPandP′areα.
SinceP′liesontheotherbranchofthehyperbola,thepolarequationoftheconic
is
Similarly,
Fromequations(9.41)and(9.42),wegetPP′=QQ′.Thatis,inaRH,perpendicularfocalchordsareofequallength.
Example9.7.7
ThetangentstoaconicatPandQmeetatT.ShowthatifSisafocusthenSTbisects∠PSQ.
Solution
Lettheequationoftheconicbe TheequationofthetangentatP
withvectorialangleαis
TheequationofthetangentatQwithvectorialangle
Atthepointofintersectionofthesetwotangents,
Example9.7.8
IfthetangentsattheextremitiesofafocalchordthroughthefocusSoftheconic
meettheaxisthroughSinTandT′showthat
Solution
LetPSQbeafocalchord.LetthevectorialanglesofPandQbeαandα+π.ThentheequationsoftangentsatPandQare
Whenthetangentsmeettheaxis,atthosepointsθ=0.
Example9.7.9
Ifachordofaconic subtendsanangle2αatthefocusthenshowthat
thelocusofthepointwhereitmeetstheinternalbisectoroftheangleis
Solution
LetPQbeachordoftheconic subtendinganangle2αatthefocus.
LettheinternalbisectorofPSQmeetsPQatT.LetthevectorialanglesofPandQbeβ−αandβ+α.LetthepolarcoordinatesofTbe(r1,β).
TheequationofthechordPQis
ThispassesthroughthepointT(r1,β).
Thelocusof(r1,β)is
Example9.7.10
ThetangentsattwopointPandQoftheconicmeetinTandPQsubtendsa
constantangle2αatthefocus.Showthat isaconstant.
Solution
Lettheequationoftheconicbe LetthevectorialanglesofPandQ
beβ−αandβ+α.
SincethepointsPandQlieontheconic,
AlsotheequationofchordPQis
PQisalsothepolarofthepointTandsoitsequationis
Identifyingequations(9.48)and(9.49),weget
Fromequations(9.46)and(9.47)and(9.50),weget
Example9.7.11
Ifafocalchordofanellipsemakesanangleαwiththemajoraxisthenshowthat
theanglebetweenthetangentsatitsextremitiesis
Solution
Lettheequationoftheconicbe
TheequationofthetangentatPis
TheequationofthetangentatQis
Transformingintocartesiancoordinatesbytakingx=rcosθ,y=sinθEquations(9.53)and(9.54)becomes,
Theslopesofthetangentsare
Ifθistheanglebetweenthetangentsthen
Theacuteanglebetweenthetangentsisgivenby
Example9.7.12
AfocalchordSPofanellipseisinclinedatanangleαtothemajoraxis.ProvethattheperpendicularfromthefocusonthetangentatPmakeswiththeaxisan
angle
Solution
Lettheequationoftheconicbe
TheequationoftangentatPis
TheequationoftheperpendicularlinetothetangentatPis
Iftheperpendicularpassesthroughthefocusthenk=0
Example9.7.13
i. IfAcirclepassingthroughthefocusofaconicwhoselatusrectumis2lmeetstheconicinfour
pointswhosedistancesfromthefocusare,r1,r2,r3,r4thenprovethat
ii. AcircleofgivenradiuspassingthroughthefocusSofagivenconicintersectsinA,b,CandD.ShowthatSA·SB·SC·SDisaconstant.
Solution
Lettheequationsoftheconicbe
Letabetheradiusofthecircleandαbetheanglethediametermakeswiththeinitialline.Thentheequationofthecircleis
Eliminatingθbetweenequations(9.59)and(9.60)wegetanequationwhoserootsarethedistancesofthepointofintersectionfromthefocus.Formequation(9.60),wegetr=2α(cosθcosα+Sinθsinα).
Fromequation(9.59),weget
Estimatingθ,weget
Dividingbyr4andrewritingtheequationinpowerof weget
Ifr1,r2,r3,r4arethedistancesofthepointsofintersectionfromthefocusthen
aretherootsoftheaboveequation.
Formequation(9.61),weget
(i.e.)SA·SB·SC·SDisaconstant.
Example9.7.14
Ifachordoftheconic subtendsaconstantangle2βatthepolethen
showthatthelocusofthefootoftheperpendicularfromthepoletothechord(e2
−sec2β)r2−elrcosθ+l2=0.
Solution
LetthevectorialanglesofPandQbeα−βandα+β.
TheequationofthechordPQis
Theequationofthelineperpendiculartothischordis
ThislinepassesthroughthefocusSandsok=0.
Fromequation(9.63),weget
Fromequation(9.65),weget
Squaringandadding(9.66)and(9.67),weget
Example9.7.15
Avariablechordofconicsubtendsaconstantangle2βatthefocusoftheconic
Showthatthechordtouchesanotherconichavingthesamefocus
anddirectrix.Showalsothatthelocusofpolesofsuchchordsoftheconicisalsoasimilarconic.
Solution
LetPQbeachordoftheconic subtendingaconstantangle2αatthe
focus.
LetT(r1,θ)bethepointofintersectionoftangentsatPandQ.ThenPQisthepolarofTandTisthepoleofPQ.LetthevectorialanglesofPandQbeα−βandα+β.ThentheequationofchordPQis
whereL=ecosβandE=esecθ.ThislineisatangenttotheconicC′.
Thisequationhasthesamefocusas Hence,theconicC′hasthesame
focusandthesameinitiallineasC.Forthegivenconic .
FortheconicC′,
∴SX=SX′.HenceX′coincideswithX.Hence,boththeconicshavethesamefocusandthesamedirectrix.Theequation
oftangentsatPandQare and
ThesetwotangentsintersectatT(r1,θ1)
Fromequations(9.68)and(9.69),weget
Substitutingθ1=αinequation(9.68),weget
Thelocusof(r1,θ1)is
Thelocusofpolesistheconichavingthesamefocusandsamedirectrixasthegivenconic.
Example9.7.16
Showthatthelocusofthepointofintersectionoftangentsattheextremitiesofavariablefocalchordisthecorrespondingdirectrix.
Solution
Lettheequationoftheconicbe
Theequationoftangentatαis
Theequationoftangentatα+πis
Let(r1,θ1)bethepointofintersectionofthesetwotangents.Then,
Addingthesetwoequations,weget
Therefore,thelocusisthecorrespondingdirectrix
Example9.7.17
Showthatthelocusofthepointofintersectionofperpendiculartangentstoaconicisacircleorastraightline.
Solution
Lettheequationoftheconicbe
LetPandQbethepointsontheconicwhosevectorialanglesareαandβ.TheequationsoftangentsatPandQare
Let(r1,θ1)bethepointofintersectionoftangentsatPandQ.Then
Fromequations(9.73)and(9.74),weget
Butα=βisnotpossible.
Formequation(9.73),weget
Expandingequations(9.70)and(9.71),weget
Sincethesetwolinesareperpendicular,wehave
Substitutingfor and ,weget
Therefore,thelocusof(r1,θ1)is(1−e2)r2+2elrcosθ−2l2=0.
Example9.7.18
Provethatpointsontheconic whosevectorialanglesareαandβ,
respectively,willbetheextremitiesofadiameterif
Solution
Theequationoftheconicis
Letα,βbetheextremitiesofadiameteroftheconic .Thenthe
tangentsatαandβareparallelandhencetheirslopesareequal.Theequationoftangentatαis
Theslopeofthistangentis−
Sincetangentsatαandβareparallel,
Example9.7.19
Ifanormalisdrawnatoneextremityofthelatusrectum,provethatthedistance
fromthefocusoftheotherpointinwhichitmeetstheconicis
Solution
Theequationoftheconicis
Theequationofthenormalat is
Solvingequations(9.75)and(9.76),wegettheirpointofintersection
Ifcosθ=0then .ThiscorrespondstothepointL.
Attheotherendofthenormal
Substitutinginequation(9.75)weget,
Example9.7.20
IfthetangentsatthepointsPandQonaconicintersectinTandthechordPQmeetsthedirectrixatRthenprovethattheangleTSRisarightangle.
Solution
LetthevectorialanglesofPandQbeαandβ,respectively.LetthetangentsatPandQmeetatT(r1,θ1).
TheequationoftangentsatPandQare and
Sincethesetangentsmeetat(r1,θ1),wehave and
θ1–α=±(θ2−β)whichimplies
LetθbethevectorialangleofR.TheequationofchordPQis
Theequationofthedirectrix
Subtracting,wegetsec
Example9.7.21
AchordPQofaconicsubtendsaconstant2γatthefocusSandtangentsatPand
QmeetinT.Provethat
Solution
Lettheequationoftheconicbe
LetthevectorialanglesofPandQbeαandβ,respectively.
Sincethesepointslieontheconic, and
IfthetangentsatPandQintersectat(r1,θ1)then
SincePQsubtendsanangle2γatS,
Exercises
1. IfPSP′andQSQ′aretwofocalchordsofaconiccuttingeachotheratrightanglesthenprovethat
=aconstant.
2. Iftwoconicshaveacommonfocusthenshowthattwooftheircommonchordspassthroughthepointofintersectionoftheirdirectrices.
3. Showthat and representthesameconic.
4. InaparabolawithfocusS,thetangentsatanypointsPandQonitmeetatT.Provethat
i. SP·SQ=ST2
ii. ThetrianglesSPTandSQTaresimilar.5. IfSbethefocus,PandQbetwopointsonaconicsuchthattheanglePSQisconstant,provethat
thelocusofthepointofintersectionofthetangentsatPandQisaconicsectionwhosefocusisS.
6. Ifthecircler+2acosθ=0cutstheconic infourrealpointsfindtheequation
inrwhichdeterminesthedistancesofthesepointsfromthepole.Also,showthatiftheiralgebraic
sumequals2aandtheeccentricityoftheconicis2cosα.
7. Provethatthetwoconics and toucheachotherif
8. P,QandRarethreepointsontheconic TangentsatQmeetsSPandSRinMandN
sothatSM=AN=lwhereSisthefocus.ProvethatthechordPQtouchestheconic.9. Provethattheportionofthetangentinterceptedbetweenthecurveanddirectrixsubtendsaright
angleatthefocus.10. Provethatthelocusofthemiddlepointsofasystemoffocalchordsofaconicsectionisaconic
sectionwhichisaparabola,ellipseorhyperbolaaccordingastheoriginalconicisaparabolaellipseorhyperbola.
11. Twoequalellipsesofeccentricityehaveonefocuscommonandareplacedwiththeiraxesatright
angles.IfPQbeacommontangentthenprovethat
12. IfthetangentsatPandQofaconicmeetatapointTandSbethefocusthenprovethatST2=SP·SQiftheconicisaparabola.
13. Aconicisdescribedhavingthesamefocusandeccentricityastheconic andthetwo
conicstouchatθ=α.Provethatthelengthofitslatusrectumis
14. Provethatthreenormalscanbedrawnfromagivenpointtoagivenparabola.Ifthenormalatα,β,
γontheconic meetatthepoint(ρ,ϕ)provethat
15. Ifthenormalsatthreepointsoftheparabola whosevectorialanglesareα,β,γmeet
inapointwhosevectorialangleisϕthenprovethat2ϕ=α+β+γ−π.
16. Ifα,β,γbethevectorialanglesofthreepointson ,thenormalatwhichare
concurrent,provethat
17. IfthenormalatPtoaconiccutstheaxisinG,provethatSN=eSP.18. IfSMandSNbeperpendicularsfromthefocusSonthetangentandnormalatanypointonthe
conic and,STtheperpendicularonMNshowthatthelocustoTisr(e2−1)=elcos
θ.
19. Showthatifthenormalsatthepointswhosevectorialanglesareθ1,θ2,θ3andθ4on
meetatthepoint(r′,ɸ)thenθ1+θ2+θ3+θ4−2ɸ=(2n+1)π.20. Provethatthechordsofarectangularhyperbolawhichsubtendarightangleatafocustoucha
fixedparabola.21. Ifthetangentatanypointofanellipsemakeanangleawithitsmajoraxisandanangleβwiththe
focalradiustothepointofcontactthenshowthatecosα=cosβ
Ans.:A2+B2−2e(Acosγ+Bsinγ)+e2−1=0
22. Provethattheexterioranglebetweenanytwotangentstoaparabolaisequaltohalfthedifferenceofthevectorialanglesoftheirpointsofcontact.
23. Findtheconditionthatthestraightline maybeatangenttotheconic
24. Findthelocusofpolesofchordswhichsubtendaconstantangleatthefocus.25. Provethatifthechordsofaconicsubtendaconstantangleatthefocus,thetangentsattheendof
thechordwillmeetonafixedconicandthechordwilltouchanotherfixedconic.
26. Findthelocusofthepointofintersectionofthetangentstotheconic atPandQ,
where ,kbeingaconstant.
27. IfthetangentandnormalatanypointPofaconicmeetthetransverseaxisofTandG,
respectively,andifSbethefocusthenprovethat isaconstant.
Chapter10
TracingofCurves
10.1GENERALEQUATIONOFTHESECONDDEGREEANDTRACINGOFACONIC
Intheearlierchapters,westudiedstandardformsofaconicnamelyaparabola,ellipseandhyperbola.Inthischapter,westudytheconditionsforthegeneralequationoftheseconddegreetorepresentthedifferenttypesofconic.Inordertostudytheseproperties,weintroducethecharacteristicsofchangeoforiginandthecoordinateaxes,rotationofaxeswithoutchangingtheoriginandreducingtheseconddegreeequationwithoutxy-term.
10.2SHIFTOFORIGINWITHOUTCHANGINGTHEDIRECTIONOFAXES
LetOxandOybetwoperpendicularlinesonaplane.LetO′beapointinthexy-plane.ThroughO′,drawO′XandO′YparalleltoOxandOy,respectively.LetthecoordinatesofO′be(h,k)withrespecttotheaxesOxandOy.DrawO′LperpendiculartoOx.ThenOL=handO′L=k.
LetP(x,y)beanypointinthexy-plane.DrawPMperpendiculartoOxmeetingaxisatN.Then
10.3ROTATIONOFAXESWITHOUTCHANGINGTHEORIGIN
LetOxandOybetheoriginalcoordinateaxes.LetOxandOyberotatedthroughanangleθintheanticlockwisedirection.
LetP(x,y)beapointinthexy-plane.DrawPLperpendiculartoOx,PM
perpendiculartoOXandMNperpendiculartoLP.Then
Let(X,Y)bethecoordinatesofthepointPwithrespecttoaxesOXandOY.Then
From(10.1)and(10.2)weseethatX=xcosθ+ysinθ,Y=−xsinθ+ycosθ.
10.4REMOVALOFXY-TERM
Herewewanttotransformtheseconddegreeequationax2+2hxy+by2+22x+2fy+c=0intoaseconddegreecurvewithoutXY-term,wheretheaxesarerotatedthroughanangleθwithoutchangingtheoriginweget
IfXY-termhastobeabsentthen
Hence,byrotatingtheaxesthroughanangleθaboutOthegeneralseconddegreeexpressionwillresultintoaseconddegreeexpressionwithoutXY-terms.
10.5INVARIANTS
Wewillnowprovethattheexpressionax2+2hxy+by2willchangetoAx2+2hXY+By2if(i)a+b=A+Band(ii)ab−h2=AB−H2.
Proof:LetP(x,y)beanypointwithrespecttoaxes(ox,oy)and(X,Y)beitscoordinateswithrespectto(OX,OY).Thenx=Xcosθ−Ysinθ,y=Xsinθ+Ycosθ.Therefore,wegetx2+y2=X2+Y2
Supposeax2+2hxy+by2=AX2+2HXY+BY2.Thenax2+2hxy+by2+λ(x2+y2)=AX2+2HXY+BY2+δ(X2+Y2).IftheLHSisoftheform(px+qy)2thenitwillbechangedintotheform[p(Xcosθ−Ysinθ)+q(Xsinθ+Ycosθ)]2=(p1X+q1Y)2.
Thiswillbeaperfectsquareifh2=(a+λ)(b+λ)
RHSwillbeaperfectsquareif
Comparing(10.3)and(10.4),wegeta+b=A+B,ab−h2=AB−H2.
10.6CONDITIONSFORTHEGENERALEQUATIONOFTHESECONDDEGREETOREPRESENTACONIC
Thegeneralequationoftheseconddegree
Iftheaxesarerotatedthroughanangleθwiththeanticlockwisedirectionthen
Thentheequationtransformedwiththeseconddegreein(X,Y)is
Now,westudyseveralcasesbasedonthevaluesofAandB.Case1:Ifab–h2=0thenA=0orB=0.SupposeA=0thentheequation(10.6)takestheform
IfG=0thenthisequationwillrepresentapairofstraightlines.IfG≠0thenwehavefrom(3).
Shiftingtheorigintothepoint theaboveequationcanbewritten
intheform whichisaparabola.
Case2:Supposeab–h2≠0thenneitherA=0norB=0.Thenequation(10.6)canbewrittenas
Shiftingtheorigintothepoint theaboveequationtakestheform
IfB=0thenequation(10.8)representsaformofstraightlinesrealorimaginary.IfK≠0thenequation(10.8)canbeexpressedintheform
whichisanellipsedependingonAandB.
IfAandBareofoppositesigns,thatis,ab−h2<0thentheequation(10.9)willrepresentahyperbola.IfB=−A,thatis,a+b=0thenequation(10.9)willrepresentarectangularhyperbola.Hencewehavethefollowingconditionforthenatureoftheseconddegreeequation(10.3)torepresentindifferentforms.Theconditionsareasfollows:
1. Itwillrepresentapairofstraightlineifabc+2fgh−af2−bg2−ch2=0.2. Itwillrepresentacircleifa=bandh=0.
3. Itwillrepresentaparabolaifab−h2=0.
4. Itwillrepresentanellipseifab−h2>0.
5. Itwillrepresentahyperbolaifab−h2<0.6. Itwillrepresentarectangularhyperbolaifa+b=0.
10.7CENTREOFTHECONICGIVENBYTHEGENERALEQUATIONOFTHESECONDDEGREE
Thegeneralequationoftheseconddegreeinxandyis
Sincethisequationhasxandyterms,thecentreisnotattheorigin.Letussupposethecentreisat(x1,y1).Letusnowshifttheoriginto(x1,y1)withoutchangingthedirectionofaxes.ThenX=x+x1,Y=y+y1.Thenequation(10.10)takestheform
Sincetheoriginisshiftedtothepoint(x1,y1)withrespecttonewaxes,thecoefficientofXandYin(10.11)shouldbezero.
Solvingthesetwoequations,weget
Thenthecoordinatesofthecentreare
Ifab−h2=0,thentheconicisaparabola.
10.8EQUATIONOFTHECONICREFERREDTOTHECENTREASORIGIN
FromtheresultobtainedinSection10.7,theequationoftheconicreferredtocentreastheoriginisax2+2hxy+by2+C1=0,where,
Hence,theequationoftheconicreferredtocentreasoriginisax2+2hxy+by2+
C1=0where
Note10.8.1:Iff=ax2+2hxy+by2+2gx+2fy+c=0then
Therefore,thecoordinatesofthecentreoftheconicaregivenbysolvingthe
equations and
Example10.8.1
Findthenatureoftheconicandfinditscentre.Alsowritedowntheequationoftheconicreferredtocentreasorigin:
i. 2x2−5xy−3y2−x−4y+5=0
ii. 5x2−6xy+5y2+22x−26y+29=0
Solution
i. Given:2x2−5xy−3y2−x−4y+5=0
Here,
Therefore,theconicisahyperbola.
Thecoordinatesofthecentrearegivenby
Solvingthesetwoequations,weget
Therefore,thecoordinatesofthecentreare Theequationoftheellipsereferredto
centreasoriginis
Therefore,theequationoftheellipsereferredtothecentreis2x2−5xy−3y+7=0.ii.
Therefore,thegivenequationrepresentsanellipse.Thecoordinatesofthecentrearegivenby
Solvingtheseequationswegetthecentreas(1,2).Theequationoftheconicreferredtocentreasoriginis5x2−6xy+5y2+C1=0whereC1=gx1+fy1+c.
C1=11×(−1)−13(2)+29=−11−26+29=−8Therefore,theequationoftheellipseis5x2−6xy+5y−8=0or5x2−6xy+5y=8.
10.9LENGTHANDPOSITIONOFTHEAXESOFTHECENTRALCONICWHOSEEQUATIONIS
ax2+2hxy+by2=1
Given:
Theequationofthecircleconcentricwiththisconicandradiusris
Homogeneousingequation(10.15)withthehelpof(10.16)weget
Thetwolinesgivenbyabovehomogeneousequationwillbeconsideredonlyiftheradiusofthecircleisequaltolengthofsemi-majoraxisorsemiminoraxis.Theconditionforthatis
Thisisaquadraticequationinr2andsoithastworootsnamely For
anellipsethevalues arebothpositiveandthelengthsofthesemi-axes
are2r1and2r2.Forahyperbolaoneofthevaluesispositiveandtheotheris
negative,thatis,say ispositiveand isnegative.Thenthelengthof
transverseaxisis2r1andthelengthofconjugateaxisis
Usingequation(10.18)inequation(10.17),weget
Thentheequationsofaxesare
Theeccentricityoftheconiccanbedeterminedfromthelengthoftheaxes.
10.10AXISANDVERTEXOFTHEPARABOLAWHOSEEQUATIONISax2+2hxy+by2+2gx+2fy+c=0
Thisequationwillrepresentaparabolaifab−h2=0.Given:
Thenequation(10.20)canbeexpressedintheform
Wechooseλsuchthatand
and
areperpendiculartoeachother.
Nowequation(10.21)canbewrittenas
Sincetheaboveequationrepresentsaparabola,theaxisoftheparabolaispx+
qy+λ=0andthetangentatthevertexis andthe
lengthofthelatusrectumis where
Example10.10.1
Tracetheconic36x2+24xy+29y2−72x+126y+81=0.
Solution
Given:
Therefore,thegivenconicrepresentsanellipse.Thecoordinatesofthecentre
aregivenby and
Solvingequation(10.25)and(10.26)wegetx=2,y=−3.Therefore,thecentreoftheellipseis(2,−3).Theequationofthisellipseinstandardformis36x2+24xy+29y2+C1=0
whereC1=gx1+fy1+c.
Thelengthoftheaxesaregivenby
Solvingforr2wegetr2=9or4∴r1=3=Lengthofthesemi-majoraxisr2=2=Lengthofthesemiminoraxis
Theequationofthemajoraxisis
Theequationofminoraxisis
Referringtothecentretheequationofmajorandminoraxesare
Themajoraxis4x+3y+1=0meetstheaxesat and
Theminoraxismeetstheaxesatthepoint and(6,0).
Theconicmeetstheaxisatpointsaregivenby
whichareimaginary.
Therefore,theconicdoesnotmeetthex-axis.Similarly,bysubstitutingy=0inequation(10.24)weget29y2+126y+81=0.
Therefore,theconicintersectsy-axisinrealpoints.
Example10.10.2
Tracetheconicx2+4xy+y2−2x+2y−6=0.
Solution
ab−h2=1×1−4=−3<0.
Therefore,theconicisahyperbola.Thecoordinatesofthecentrearegivenby
(i.e.)2x+4y−2=0orx+2y−1=0
4x+2y+2=0or2x+y+1=0.Solvingthesetwoequationswegetthe
centre.Therefore,thecoordinatesofthecentreare(−1,1).Theequationoftheconicreferredtothecentreasoriginwithoutchangingthe
directionsoftheaxisisx2+4xy+y2+C1=0whereC1=gx+fy+c
Therefore,thelengthsoftheaxesaregivenby
Hence,thesemi-transverseaxisis
Thelengthofsemi-conjugateaxisis
Semi-latusrectum
Theequationofthetransverseaxisis
Theequationofconjugateaxesis
(i.e.)x+y=0
Thepointswherethehyperbolameetsthex-axisaregivenbyx2−2x−6=0.
Whenthecurvemeetsthey-axis,x=0∴y2+2y−6=0∴y=1.7or−3.7nearlyHence,thecurvepassesthroughthepoints(−1.7,0),(3.7,0),(0,1.7)and(0,
−3.7).Thecurveistracedinfollowingfigure:
Example10.10.3
Tracetheconicx2+2xy+y2−2x−1=0.
Solution
a=1,b=1,h=1,g=−1,f=0,c=−1Here,h2=abandabc+2fgh–af2−bg2−ch2≠0.Therefore,theconicisaparabola.Thegivenequationcanbewrittenas(x+y)2=2x+1.Theequationcanbewrittenas
whereλischosensuchthatx+y+λ=0and2(λ+1)x+2λy+(λ2+1)=0areperpendicular.
Nowequation(10.27)canbewrittenas
whichisoftheformy2=4ax.
Therefore,lengthsoflatusrectumoftheparabolais
Theaxisoftheparabolais
or2x+2y−1=0
Theequationofthetangentatthevertexis
Vertexoftheparabolais
Whentheparabolameetsthex-axis,y=0.
x2−2x−1=0∴x=2.4,−0.4Therefore,thepointsonthecurveare(−0.4,0)and(2.4,0).Whenthecurvemeetsthey-axis,x=0.
y2=1ory=±1Hence(0,−1)and(0,1)arepointsonthecurve.Thegraphofthecurveisgivenasfollows:
Exercises
Tracethefollowingconics:
1. 9x2+24xy+16y2−44x+108y−124=0
2. 5x2−6xy+5y2+22x−26y+29=0
3. 32x2+52xy−7y2−64x−52y−148=0
4. x2+24xy+16y2−86x+52y−139=0
5. 43x2+48xy+57y2+10x+180y+25=0
6. x2−4xy+4y2−6x−8y+1=0
7. x2+2xy+y2−4x−y+4=0
8. 5x2−2xy+5y2+2x−10y−7=0
9. 22x2−12xy+17y2−112x+92y+178=0.
Chapter11
ThreeDimension
11.1RECTANGULARCOORDINATEAXES
Tolocateapointinaplane,twonumbersarenecessary.Weknowthatanypointinthexyplanecanberepresentedasanorderedpair(a,b)ofrealnumberswhereaiscalledx-coordinateofthepointandbiscalledthey-coordinateofthepoint.Forthisreason,aplaneiscalledtwodimensional.Tolocateapointinspace,threenumbersarerequired.Anypointinspaceis
representedbyanorderedtriple(a,b,c)ofrealnumbers.Torepresentapointinspacewefirstchooseafixedpoint‘O’(calledtheorigin)andthenthreedirectedlinesthroughOwhichareperpendiculartoeachother(calledcoordinateaxes)andlabelledx-axis,y-axisasbeinghorizontalandz-axisasverticalandwetaketheorientationoftheaxes.Inordertodothis,wefirstchooseafixedpointO.Inlookingatthefigure,youcanthinkofy-andz-axesaslyingintheplaneofthepaperandx-axisascomingoutofthepapertowardsy-axis.Thedirectionofz-axisisdeterminedbytheneighbourhoodrule.Ifyoucurlthefingersofyourright-handaroundthez-axisinthedirectionofa90°counterclockwiserotationfromthepositivex-axistothepositivey-axisthenyourthumbpointsinthepositivedirectionofthez-axis.
Thethreecoordinateaxesaredeterminedbythethreecoordinateplanes.Thexy-planeistheplanethatcontainsxandy-axes,theyz-planecontainsyandz-axesandthexz-planecontainsx-andz-axes.
Thesethreecoordinateplanesdividethespaceintoeightpartscalledoctants.Thefirstoctantisdeterminedbythepositiveaxes.Lookatanybottomcornerofaroomandcallthecornerasorigin.
Thewallonyourleftisinthexz-plane.Thewallonyourrightisintheyz-plane.Thewallonthefloorisinthexy-plane.Thex-axisrunsalongthe
intersectionofthefloorandtheleftwall.They-axisrunsalongtheintersectionofthefloorandtherightwall.Thez-axisrunsupfromthefloortowardstheceilingalongtheintersectionofthetwowallsaresituatedinthefirstoctantandyoucannowimaginesevenotherroomssituatedintheothersevenoctants(threeonthesamefloorandfourthonthefloorbelow),allareconnectedbythecommonpointO.IfPisanypointinspace,andabethedirecteddistanceinthefirstoctant
fromtheyz-planetoP.Letthedirecteddistancefromthexz-planebebandletcbethedistancefromxy-planetoP.
WerepresentthepointPbytheorderedtriple(a,b,c)ofrealnumbersandwecalla,bandcthecoordinatesofP.aisthex-coordinate,bisthey-coordinateandcisthez-coordinate.Thus,tolocatethepoint(a,b,c)wecanstartattheoriginOandmoveaunitsalongx-axisthenbunitsparalleltoy-axisandcunitsparalleltothez-axisasshownintheabovefigure.
ThepointP(a,b,0)determinesarectangularboxasintheabovefigure.IfwedropaperpendicularfromPtothexy-planewegetapointC′withcoordinatesP(a,b,0)calledtheprojectionofPonthexy-plane.Similarly,B′(a,0,c)and
A′(0,b,c)aretheprojectionsofPonxz-planeandyz-plane,respectively.ThesetofallorderedtriplesofrealnumbersistheCartesianproductR×R×R={(x,y,z)|x,y,z∈R},whichisR3.Wehaveaone-to-onecorrespondencebetweenthepointsPinspaceandorderedtriples(a,b,c)inR3.Itiscalledathree-dimensionalrectangularcoordinatesystem.Wenoticethatintermsofcoordinates,thefirstoctantcanbedescribedastheset{(x,y,z)|x≥0,y≥0,z≥0}.
11.2FORMULAFORDISTANCEBETWEENTWOPOINTS
Considerarectangularbox,wherePandQaretheoppositecornersandthefacesoftheboxareparalleltothecoordinateplanes.IfA(x1,y1,z1)andB(x2,y2,z2)aretheverticesoftheboxindicatedintheabovefigure,then
|PA|=|x2–x1|,|AB|=|y2–y1|,BQ=|z2–z1|SincethetrianglesPBQandPABarebothrightangled,byPythagoras
theorem,
Example11.2.1
Findthedistancebetweenthepoints(2,1,–5)and(4,–7,6).
Solution
Thedistancebetweenthepoints(2,1,–5)and(4,–7,6)is
units
Aliter:LetP(x1,y1,z1)and(x2,y2,z2)betwopoints.DrawPA,QBperpendiculartoxy-plane.ThenthecoordinatesofAandBare(x1,y1,0)and(x2,y2,0).
(i.e.)(x1,y1)and(x2,y2)inthexy-plane.
∴AB2=(x2–x1)2+(y2–y1)2
DrawPCperpendiculartoxy-plane.PAandPBbeingperpendiculartoxy-plane,PAandQBarealsoperpendiculartoAB.
∴PABCisarectangleandsoPC=ABandPA=CB.
Fromtriangle
Example11.2.2
IfOistheoriginandPisthepoint(x,y,z)thenOP2=x2+y2+z2or
11.2.1SectionFormula
Thecoordinatesofapointthatdividesthelinejoiningthepoints(x1,y1,z1)and(x2,y2,z2)areintheratiol:m.LetR(x,y,z)dividethelinejoiningthepointsP(x1,y1,z1)andQ(x2,y2,z2)in
theratiol:m.
DrawPL,QCandRBperpendiculartothexy-plane.ThenACBisastraightlinesinceprojectionofastraightlineonaplaneisastraightline.ThroughRdrawastraightlineLRMparalleltoACBtomeetAP(produced)inAandCQinM.ThentrianglesLPRandMRQaresimilar.
However,
Therefore,from(11.1),
Similarly,bydrawingperpendicularsonxzandyzplaneswecanprovethat
Therefore,thecoordinatesofRare
Note11.2.1.1:IfR′dividesPQexternallyintheratiol:mthen,
∴Therefore,changeminto–mtogetthecoordinatesofR′,theexternalpointofdivision.Thecoordinatesofexternalpointofdivisionare
Note11.2.1.2:TofindthemidpointofPQtakel:m=1:1.
Thenthecoordinatesofmidpointare
11.3CENTROIDOFTRIANGLE
LetABCbeatrianglewithverticesA(x1,y1,z1),B(x2,y2,z2)andC(x3,y3,z3).
ThenthemidpointofBCisD
LetGbethecentroidofthetriangleABC.ThenGdividesADintheratio2:1.ThentheCoordinatesofGare
Hence,thecentroidofΔABCis
11.4CENTROIDOFTETRAHEDRON
LetOBCDbeatetrahedronwithvertices(xi,yi,zi),i=1,2,3,4.
ThecentroidofthetetrahedrondividesADintheratio3:1.Therefore,thecoordinatesofGare
11.5DIRECTIONCOSINES
Directioncosinesinthree-dimensionalcoordinategeometryplayarolesimilartoslopeintwo-dimensionalcoordinategeometry.
Definition11.5.1:Ifastraightlinemakesanglesα,βandγwiththepositivedirectionsofx-,y-andz-axesthencosα,cosβandcosγarecalledthedirectioncosinesoftheline.Thedirectionalcosinesaredenotedbyl,mandn.
∴l=cosα,m=cosβ,n=cosγ.Thedirectioncosinesofx-axisare1,0and0.Thedirectioncosinesofy-axisare0,1and0.Thedirectioncosinesofz-axisare0,0and1.
IfOistheoriginandP(x,y,z)beanypointinspaceandOP=r,thenthedirectioncosinesofthelinearelr,mr,nr.LetObetheoriginandP(x,y,z)isanypointinspace.DrawPNperpendiculartoXOYplane.DrawNL,NMparalleltoy-andx-axes.
ThenOL=x1,OM=y1,PN=z1.
Also,
Similarly,x=rcosαandz=rcosγ.ThenthecoordinatesofPare
Note11.5.2:ThedirectioncosinesofthelineOPare
Note11.5.3:IfOP=1unitthenthedirectioncosinesofthelineare(x,y,z).Thatis,thecoordinatesofthepointParethesameasthedirectioncosinesofthelineOP.
Note11.5.4:IfOP=1unitandPisthepoint(x,y,z)thenOP2=x2+y2+z2orx2+y2+z2=1.
∴l2+m2+n2=1Therefore,directioncosinessatisfythepropertycos2α+cos2β+cos2γ=1.
11.5.1DirectionRatios
Supposea,bandcarethreenumbersproportionaltol1,m1andn1(thedirectioncosinesofaline),then
Therefore,thedirectioncosinesofthelineare
wherethesamesignistakenthroughout.Herea,band
carecalledthedirectionratiosoftheline.If2,3and5arethedirectionratiosofalinethenthedirectioncosinesofthelineare
11.5.2ProjectionofaLine
TheprojectionofalinesegmentABonalineCDisthelinejoiningthefeetoftheperpendicularsfromAandBonCD.IfALmakesanangleθwiththelineCD
then whereALisparalleltoCD.
Therefore,theprojectionofABonCDisLM=ABcosθ.
Note11.5.2.1:TheprojectionofbrokenlinesAB,BCandCDonthelineCDisLM,MNandND.
∴Therefore,thesumoftheprojectionAB,BCandCDisLM+MN+ND=LP.
11.5.3DirectionCosinesoftheLineJoiningTwoGivenPoints
LetP(x1,y1,z1)andQ(x2,y2,z2)bethegivenpoints.WeeasilyseethattheprojectionofPQonx-,y-andz-axesarex2–x1,y2–y1andz2–z1.However,theprojectionsofPQonx-,y-andz-axesarealsoPQcosα,PQcosβandPQcosγ.
Inaddition, SincePQisofconstantlength,thedirectionratios
ofPQare
(x2–x1,y2–y1,z2−z1).
11.5.4AnglebetweenTwoGivenLines
Let(l1,m1,n1),(l2,m2,n2)linesnamelyABandCD.ThroughbethedirectioncosinesofthetwogivenOdrawlinesparalleltoABandCDandtakepointsPandQsuchthatOP=OQ=1unit.
SinceOPandOQareparalleltothetwogivenlinesthentheanglebetweenthetwogivenlinesisequaltotheanglebetweenthelinesOPandOQ.SinceOP=OQ=1unit,thecoordinatesofPandQare(l1,m1,n1)and(l2,m2,n2).Let
ThenPQ2=OP2+OQ2–2·OP·OQcosθ=1+1−2·1·1cosθ
Also,
From(11.2)and(11.3),
Note11.5.4.1:Ifthetwolinesareperpendicularthenθ=90°andcos90°=0.
∴from(11.3),l1l2+m1m2+n1n2=0
Note11.5.4.2:
Note11.5.4.3:
Note11.5.4.4:Ifa1,b1,c1anda2,b2,c2arethedirectionratiosofthetwolinesthen
Ifthetwolinesareparallelthensinθ=0.
Also,ifa1,b1,c1anda2,b2,c2arethedirectionratiosoftwoparallellinesthen
ILLUSTRATIVEEXAMPLES
Example11.1
Showthatthepoints(–2,5,8),(–6,7,4)and(–3,4,4)formaright-angledtriangle.
Solution
ThegivenpointsareA(–2,5,8),B(–6,7,4)andC(−3,4,4).
SinceBC2+AC2=AB2,thetriangleisrightangled.SinceBC=AC,thetriangleisalsoisosceles.
Example11.2
Showthatthepoints(3,2,5),(2,1,3),(–1,2,1)and(0,3,3)takeninorderformaparallelogram.
Solution
LetthefourpointsbeA(3,2,5),B(2,1,3),C(–1,2,1)andD(0,3,3).Then,
SinceAB=CDandBC=AD,thefourpointsformaparallelogram.
Aliter:ThemidpointofACis(1,2,3).ThemidpointofBDis(1,2,3).Therefore,inthefigureABCD,thediagonalsbisecteachother.HenceABCD
isaparallelogram.
Example11.3
Showthatthepoints(–1,2,5),(1,2,3)and(3,2,1)arecollinear.
Solution
ThethreegivenpointsareA(–1,2,5),B(1,2,3)andC(3,2,1).
Hence,thethreegivenpointsarecollinear.
Example11.4
Showthatthepoints(3,2,2),(–1,1,3),(0,5,6)and(2,1,2)lieonaspherewhosecentreis(1,3,4).Alsofinditsradius.
Solution
LetthegivenpointsbeS(2,1,2),P(3,2,2),Q(–1,1,3),R(0,5,6)andC(1,3,4).
Therefore,thepointsP,Q,RandSlieonaspherewhosecentreisC(1,3,4)andwhoseradiusis3units.
Example11.5
Findtheratioinwhichthestraightlinejoiningthepoints(1,–3,5)and(7,2,3)isdividedbythecoordinateplanes.
Solution
LetthelinejoiningthepointsP(1,–3,5)andQ(7,2,3)bedividedbyXY,YZandZXplanesintheratiol:1,m:1andn:1,respectively.WhenthelinePQmeetstheXYplanes,theZ-coordinatesofthepointofmeetis0.
(i.e.)TheratioinwhichPQdividestheplaneYZ-planeis1:7externally.
Similarly,
SinceXZ-planedividesPQintheratio3:2internally.
Also or Therefore,XY-planedividesPQintheratio5:3
externally.
Example11.6
PandQarethepoints(3,4,12)and(1,2,2).FindthecoordinatesofthepointsinwhichthebisectoroftheanglePOQmeetsPQ.
Solution
Weknowthat
RdividesPQinternallyintheratio13:3andSdividesPQexternallyintheratio13:3.
Therefore,thecoordinatesofRare
(i.e.)
(i.e.)
SdividesPQexternallyintheratio13:3.
Therefore,thecoordinatesofSare
(i.e.)
Example11.7
Provethatthethreelineswhichjointhemidpointsofoppositeedgesofatetrahedronpassthroughthesamepointandarebisectedatthatpoint.
Solution
LetABCDbeatetrahedronwithvertices(xi,yi,zi),i=1,2,3,4.Thethreepairsofoppositeedgesare(AD,BC),(BD,AC)and(CD,AB).Let(L,N),(P,Q)and(R,S)bethemidpointsofthethreepairsofoppositeedges.ThenListhepoint
Misthepoint
ThemidpointofLMis
Bysymmetry,thisisalsothemidpointofthelinesPQandRS.Therefore,thelinesLM,PQandRSareconcernedandarebisectedatthat
point.
Example11.8
Aplanetriangleofsidesa,bandcisplacedsothatthemidpointsofthesidesareontheaxes.Showthatthelengthsl,mandninterceptedontheaxesaregivenby8l2=b2+c2–a2,8m2=c2+a2–b2and8n2=a2+b2–c2andthatthecoordinatesoftheverticesofthetriangleare(–l,m,n),(l,–m,n)and(l,m,–n).
Solution
LetD,EandFbethemidpointsofthesidesBC,CAandAB,respectively.D,EandFarethepoints(l,0,0),(0,m,0),(0,0,n),respectively.LetA,BandCbethepoints(x1,y1,z1),(x2,y2,z2)and(x3,y3,z3),respectively.Then,
Similarly,
Therefore,theverticesare(–l,m,n),(l,–m,n)and(l,m,–n).
Example11.9
Adirectedlinemakesangles60°and60°withx-andy-axes,respectively.Findtheangleitmakeswithz-axis.
Solution
Ifalinemakesanglesα,βandγwithx-,y-andz-axes,respectivelythencos2α+cos2β+cos2γ=1.
Here,α=60°,β=60°
Example11.10
Findtheacuteanglebetweenthelineswhosedirectionratiosare2,1,–2and1,1,0.
Solution
Thedirectioncosinesofthetwolinesare
Ifθistheanglebetweenthelinesthen
Example11.11
Findtheanglebetweenanytwodiagonalsofaunitcube.
Solution
ThefourdiagonalsofthecubeareOO′,AA′,BB′andCC′.ThenthedirectionratiosofOO′andAA′are(1,1,1)and(−1,1,1).ThedirectioncosinesofOO′andAA′
are and Ifθistheanglebetweenthesetwodiagonals
then
Similarly,theanglebetweenanytwodiagonalsis
Example11.12
Ifα,β,γandδaretheanglesmadebyalinewiththefourdiagonalsofacube,
provethatcos2α+cos2β+cos2γ+cos2
Solution
ThefourdiagonalsareOO′,AA′,BB′andCC′(referfiguregiveninExample11.11).Letl,m,nbethedirectioncosinesofthelinemakinganglesα,β,γandδwiththefourdiagonals.Then,
Squaringandaddingthesefourresults,weget
Example11.13
Ifl1,m1,n1andl2,m2,n2arethedirectioncosinesoftwomutuallyperpendicularlines,showthatthedirectioncosinesofthelinesperpendiculartotheabovetwolinesare
m1n2–m2n1,n1l2−l1n2andl1m2–l2m1.
Solution
Letl,mandnbethedirectioncosinesofthelineperpendiculartothetwogivenlines.Thenll1+mm1+nn1=0;ll2,+mm2+nn2=0
But andsincethetwolinesareperpendicular
Therefore,thedirectioncosinesofthelineperpendiculartothegiventwolinesarem1n2–m2n1,n1l2–n2l1,l1m2–l2m1.
Example11.14
Showthatthreeconcurrentstraightlineswithdirectioncosinesl1,m1,n1;l2,m2,n2andl3,m3,n3arecoplanarif
Solution
Letl,mandnbethedirectioncosinesofthelinewhichisperpendiculartothegiventhreelines.Ifthelinesarecoplanarthenthelinewithdirectioncosinesl,mandnisnormaltothegivencoplanarline.
Eliminatingl,mandnweget
Example11.15
Provethatthestraightlineswhosedirectioncosinesaregivenbytheequations
al+bm+cn=0andfmn+gnl+hlm=0areperpendicularif
Solution
Thedirectioncosinesoftwolinesaregivenby
From(11.5),
Substitutingin(11.6),weget
Dividingbym2,weget|
Ifl1,m1,n1andl2,m2,n2arethedirectioncosinesofthetwogivenlinesthen
and aretherootsoftheequation(11.7),then
Similarly,
Butl1l2+m1m2+n1n2=0
Dividing
Example11.16
Iftwopairofoppositeedgesofatetrahedronareatrightanglesthenshowthatthethirdpairisalsoatrightangles.
Solution
Let(OA,BC),(OB,CA)and(OC,AB)bethreepairofoppositeedges.LetObetheorigin.LetthecoordinatesofA,BandCbe(x1,y1,z1),(x2,y2,z2)and(x3,y3,z3),respectively.ThenthedirectionratiosofOAandBCarex1,y1,z1andx2–x3,y2–y3,z2–z3.SinceOAisperpendiculartoBC,weget
SinceOBisperpendiculartoACweget
Adding(11.8)and(11.9),weget
ThisshowsthatOCisperpendiculartoAB.
Example11.17
Ifl1,m1,n1;l2,m2,n2andl3,m3,n3bethedirectioncosinesofthemutuallyperpendicularlinesthenshowthatthelinewhosedirectionratiosl1+l2+l3,m1+m2+m3andn1+n2+n3makeequalangleswiththem.
Solution
Ifl1,m1,n1;l2,m2,n2andl3,m3,n3arethedirectioncosinesofthreemutually
perpendicularlines also
Letθbetheanglebetweenthelineswiththedirectioncosinesl1,m1,n1anddirectionratiosl1+l2+l3,m1+m2+m3andn1+n2+n3.Then,
Similarly,theothertwoanglesareequaltothesamevalueofθ.Therefore,thelineswiththedirectionratiosl1+l2+l3,m1+m2+m3,n1+n2+n3areequallyinclinedtothelinewithdirectioncosinesl1,m1,n1;l2,m2,n2and
l3,m3,n3.
Example11.18
Showthatthestraightlineswhosedirectioncosinesaregivenbya2l+b2m+c2n=0,mn+nl+lm=0willbeparallelifa±b±c=0.
Solution
Giventhedirectioncosinesoftwogivenlinessatisfytheequations
From(11.11),
Substitutingthisvalueofnin(11.12),weget
Dividingby
Ifl1,m1,n1andl2,m2,n2arethedirectioncosinesofthetwogivenlinesthen
aretherootsoftheequation(11.13).
Also,ifthelinesareparallelthen thentherootsoftheequation(11.13)
areequal.Theconditionforthatisthediscriminantisequaltozero.
Example11.19
Theprojectionsofalineontheaxesare3,4,12.Findthelengthanddirectioncosinesoftheline.
Solution
Let(l,m,n)bethedirectioncosinesofthelineand(x1,y1,z1)and(x2,y2,z2)betheextremitiesoftheline.Thedirectioncosinesofx-,yandz-axesare(1,0,0),(0,1,0)and(0,0,1),respectively.Theprojectionofthelineontheaxisis3.
∴x2–x1=3.Similarly,y2–y1=4,z2–z1=12
Thendirectionratiosofthelineare3,4,12.
Therefore,thedirectioncosinesofthelineare
Exercises
1. Showthatthepoints(10,7,0),(6,6,–1)and(6,9,–4)formanisoscelesright-angledtriangle.2. Showthatthepoints(2,3,5),(7,5,–1)and(4,–3,2)formanisoscelestriangle.3. Showthatthepoints(1,2,3),(2,3,1)and(3,1,2)formanequilateraltriangle.
4. Showthatthepoints(4,0,5),(2,1,3)and(1,3,2)arecollinear.5. Showthatthepoints(1,–1,1),(5,–5,4),(5,0,8)and(1,4,5)formarhombus.6. Provethatthepoints(2,–1,0),(0,–1,–1),(1,1,–3)and(3,1,–2)formtheverticesofarectangle.7. Showthatthepoints(1,2,3),(–1,2,–1),(2,3,2)and(4,7,6)formaparallelogram.8. Showthatthepoints(–4,3,6),(–5,2,2),(–8,5,2),(–7,6,6)formarhombus.9. Showthatthepoints(4,–1,2),(0,–2,3),(1,–5,–1)and(2,0,1)lieonaspherewhosecentreis
(2,–3,1)andfinditsradius.10. Findtheratioinwhichthelinejoiningpoints(2,4,5)and(3,5,–4)isdividedbythexy-plane.
Ans.:(5,4)
11. ThelinejoiningthepointsA(–2,6,4)andB(1,3,7)meetstheYOZ-planeatC.FindthecoordinatesofC.
Ans.:(0,4,6)
12. ThreeverticesofaparallelogramABCDareA(3,–4,7),B(–5,3,–2)andC(1,2,–3).FindthecoordinatesofD.
Ans.:(9,–5,6)
13. Showthatthepoints(–5,6,8),(1,8,11),(4,2,9)and(–2,0,6)aretheverticesofasquare.14. ShowthatthepointsP(3,2,–4),Q(9,8,–10)andR(5,4,–6)arecollinear.Findtheratioinwhich
RdividesPQ.Ans.:(1,2)
15. Findtheratioinwhichthecoordinateplanesdividethelinejoiningthepoints(–2,4,7)and(3,–5,8).
Ans.:7:9;4:5;–7:–8
16. Provethatthelinedrawnfromtheverticesofatetrahedrontothecentroidsoftheoppositefacesmeetinapointwhichdividesthemintheratio3:1.
17. Findthecoordinateofthecircumcentreofthetriangleformedbythepointswithvertices(1,2,1),(–2,2,–1)and(1,1,0).
Ans.:
18. AandBarethepoints(2,3,5)and(7,2,4).FindthecoordinatesofthepointswhichthebisectorsoftheanglesAOBmeetAB.
19. FindthelengthofthemedianthroughAofthetriangleA(2,–1,4),B(3,7,–6)andC(–5,0,2).Ans.:7
20. Provethatthelocusofapoint,thesumofwhosedistancesfromthepoints(a,0,0)and(–a,0,0)
isaconstant2k,isthecurve
21. Whatarethedirectioncosinesofthelinewhichisequallyinclinedtotheaxes?
Ans.:
22. Findtheanglebetweenthelineswhosedirectionratiosare(2,3,4)and(1,–2,1).
Ans.:
23. Avariablelineintwoadjacentpositionshasdirectioncosines(l,m,n),(l+δl,m+δm,n+δn).
Provethatthesmallangleδθbetweentwopositivesisgivenbyδ2θ=(δl)2+(δm)2+(δn)2.24. FindtheanglebetweenthelinesABandCD,whereA,B,CandDarethepoints(3,4,5),(4,6,3),
(–1,2,4)and(1,0,5),respectively.
Ans.:
25. Provebydirectioncosinesthepoints(1,–2,3),(2,3,–4)and(0,–7,10)arecollinear.26. Findtheanglebetweenthelineswhosedirectionratiosare(2,1,–2)and(1,–1,0).
Ans.:
27. Showthatthelinejoiningthepoints(1,2,3)and(1,5,7)isparalleltothelinejoiningthepoints(–4,3,–6)and(2,9,2).
28. P,Q,RandSarethepoints(2,3,–1),(3,5,3),(1,2,3)and(2,5,7).ShowthatPQisperpendiculartoRS.
29. Provethatthethreelineswithdirectionratios(1,–1,1),(1,–3,0)and(1,0,3)lieinaplane.
30. Showthatthelineswhosedirectioncosinesaregivenbyal+bm+cn=0andal2+bm2+cn2=0
areparallelif
31. Showthatthelineswhosedirectioncosinesaregivenbytheequationsal+vm+wn=0andal2+
bm2+cn2=0areparallelifu2(b+c)+v2(c+a)+w2(a+b)=0andperpendicularif
32. Iftheedgesofarectangularparallelepipedarea,bandc,showthattheanglebetweenthefour
diagonalsaregivenbycos-1
33. Ifinatetrahedronthesumofthesquaresofoppositeedgesisequal,showthatitspairsofoppositesidesareatrightangles.
34. Findtheanglebetweenthelineswhosedirectioncosinesaregivenbytheequations:
i. l+m+n=0andl2+m2–n2=0ii. l+m+n=0and2lm–2nl–mn=0
Ans.:(i)
(ii)
35. If(l1,m1,n1)and(l2,m2,n2)arethedirectioncosinesoftwolinesinclinedatanangleq,showthattheactualdirectioncosinesofthedirectionbetweenthelinesare
36. AB,BCarethediagonalsofadjacentfacesofarectangularboxwithcentreattheoriginOitsedgesbeingparalleltoaxes.IftheanglesAOB,BOCandCOAareθ,ϕandω,respectivelythenprovethatcosθ+cosϕ+cosω=−1.
37. Iftheprojectionsofalineontheaxesare2,3,6thenfindthelengthoftheline.Ans.:7
38. ThedistancebetweenthepointsPandQandthelengthsoftheprojectionsofPQonthe
coordinateplanesared1,d2andd3,showthat
39. Showthatthethreelinesthroughtheoriginwithdirectionratios(1,–1,7),(1,–1,0)and(1,0,3)lieonaplane.
40. Showthattheanglebetweenthelineswhosedirectioncosinesaregivenbyl+m+n=0andfmn
+gnl+hlm=0is
Chapter12
Plane
12.1INTRODUCTION
Inthree-dimensionalcoordinategeometry,firstwedefineaplaneandfromaplanewedefineastraightline.Inthischapter,wedefineaplaneandobtainitsequationindifferentforms.Wealsoderiveformulatofindtheperpendiculardistancefromagivenpointtoaplane.Also,wefindtheratioinwhichaplanedividesthelinejoiningtwogivenpoints.
Definition12.1.1:Aplaneisdefinedtobeasurfacesuchthatthelinejoininganytwopointswhollyliesonthesurface.
12.2GENERALEQUATIONOFAPLANE
Everyfirstdegreeequationinx,yandzrepresentsaplane.Considerthefirstdegreeequationinx,yandzas
wherea,b,canddareconstants.LetP(x1,y1,z1)andQ(x2,y2,z2)betwopointsonthelocusofequation(12.1).Thenthecoordinatesofthepointsthatdivide
linejoiningthesetwopointsintheratioλ:1are If
thispointliesonthelocusofequation(12.1)then
SinceP(x1,y1,z1)andQ(x2,y2,z2)aretwopointsonthelocusoftheequation(12.1)thesetwopointshavetosatisfythelocusoftheequation(12.1).
Multiplying(12.4)byλandaddingwith(12.3),weget(ax1+by1+cz1+d)+λ(ax2+by2+cz2+d)=0whichistheequation(12.2).
Therefore,thepoint liesonthelocusofequation
(12.1).Hence,thisshowsthatiftwopointsliesonthelocusofequation(12.1)then
everypointonthislineisalsoapointonthelocusofequation(12.1).Hence,theequation(12.1)representsaplaneandthuswehaveshownthateveryfirstdegreeequationinx,yandzrepresentsaplane.
12.3GENERALEQUATIONOFAPLANEPASSINGTHROUGHAGIVENPOINT
Lettheequationoftheplanepassingthroughagivenpoint(x1,y1,z1)be
since(x1,y1,z1)liesontheplane(12.5).
Subtracting(12.6)from(12.5),wegeta(x–x1)+b(y–y1)+c(z–z1)=0.Thisisthegeneralequationoftheplanepassingthroughthegivenpoint(x1,y1,z1).
12.4EQUATIONOFAPLANEININTERCEPTFORM
Lettheequationofaplanebe
Letthisplanemakeinterceptsa,bandcontheaxesofcoordinates.Ifthisplanemeetsthex-,y-andz-axesatA,BandCthentheircoordinatesare(a,0,0),(0,b,0)and(0,0,c),respectively.SincethesepointslieontheplaneAx+By+Cz+D=0,thecoordinatesofthepointshavetosatisfytheequationAx+By+Cz+D=0.
ByreplacingthevaluesofA,BandC,weget
Thisequationiscalledtheinterceptformofaplane.
12.5EQUATIONOFAPLANEINNORMALFORM
LetaplanemeetthecoordinateaxesatA,BandC.DrawONperpendiculartotheplaneABCandletON=p.LetthedirectioncosinesofONbe(cosα,cosβ,cosγ).SinceON=p,thecoordinatesofNare(pcosα,pcosβ,pcosγ).Letp(x1,y1,z1)beanypointintheplaneABC.Ifalineisperpendiculartoaplanethenitisperpendiculartoeverylinetotheplane.Therefore,ONisperpendiculartoOP.SincethecoordinatesofPandNare(x1,y1,z1)and(pcosα,pcosβ,pcosγ)thedirectionratiosofNare(x1–pcosα,y1–pcosβ,z1–pcosγ)sinceNisperpendiculartoON.
Therefore,thelocusof(x1,y1,z1)isxcosα+ycosβ+zcosγ=p.Thisequationiscalledthenormalformofaplane.
Note12.5.1:Here,thecoefficientsofx,yandzarethedirectioncosinesofnormaltotheplaneandpistheperpendiculardistancefromtheoriginontheplane.
Note12.5.2:Reductionofaplanetonormalform:theequationofplaneingeneralformis
Itsequationinnormalformis
Identifying(12.8)and(12.9),weget
SincephastobepositivewhenDispositive,wehave
12.6ANGLEBETWEENTWOPLANES
Lettheequationoftwoplanesbe
Thedirectionratiosofthenormalstotheaboveplanesare
Theanglebetweentwoplanesisdefinedtobetheanglebetweenthenormalstothetwoplanes.Letθbetheanglebetweentheplanes.
Note12.6.1:Thepositivesignofcosθgivestheacuteanglebetweentheplanesandnegativesigngivestheobtuseanglebetweentheplanes.
Note12.6.2:Iftheplanesareperpendicularthenθ=90°.
∴a1a2+b1b2+c1c2=0
Note12.6.3:Iftheplanesareparallelthendirectioncosinesofthenormalsareproportional.
Note12.6.4:Theequationofplaneparalleltoax+by+cz+d=0canbeexpressedintheformax+by+cz+k=0.
12.7PERPENDICULARDISTANCEFROMAPOINTONAPLANE
Lettheequationoftheplanebe
andP(x1,y1,z1)bethegivenpoint.WehavetofindtheperpendiculardistancefromPtotheplane.Thenormalformoftheplane(12.12)is
DrawPMperpendicularfromPtotheplane(12.12).DrawtheplanethroughPtothegivenplane(12.12).Theequationofthisplaneis
wherep1istheperpendiculardistancefromtheorigintotheplane(12.13).Thisplanepassesthrough(x1,y1,z1).
∴lx1+my1+nz1=p1DrawBNperpendiculartotheplane(12.3)meetingtheplane(12.12)atM.
ThenON=p1andOM=p.
MN=OM–ON=p–p1=p–(lx1+my1+nz1).Comparingequations(12.12)and(12.13),
Therefore,theperpendiculardistancefrom(x1,y1,z1)is= .
Aliter:
LetPMbetheperpendicularfromPontheplaneax+by+cz+d=0.LetP(x1,y1,z1)andM(x2,y2,z2)beapointontheplane(12.12).ThenQMandPMare
perpendicular.Letθbethe ThedirectionratiosofPMandPNare(a,b,c)
and(x1–x2,y1–y2,z1–z2).
Note12.7.1:Theperpendiculardistancefromtheorigintotheplaneax+by+cz
+d=0is .
12.8PLANEPASSINGTHROUGHTHREEGIVENPOINTS
Let(x1,y1,z1),(x2,y2,z2)and(x3,y3,z3)bethreegivenpointsonaplane.Lettheequationoftheplanebe
Anyplanethrough(x1,y1,z1)is
Thisplanealsopassesthrough(x2,y2,z2)and(x3,y3,z3).
Eliminatinga,bandcfrom(12.6),(12.7)and(12.18),weget
Thisistheequationoftherequiredplane.
Aliter:Lettheequationoftheplanebe
Thisplanepassesthroughthepoints(x1,y1,z1),(x2,y2,z2)and(x3,y3,z3).
Eliminatinga,b,andcfrom(12.20),(12.21)and(12.22),weget
Thisistheequationoftherequiredplane.
12.9TOFINDTHERATIOINWHICHTHEPLANEJOININGTHEPOINTS(x1,y1,z1)AND(x2,y2,z2)ISDIVIDEDBYTHEPLANEax+by+cz+d=0.
Theequationoftheplaneis
LetthelinejoiningthepointsP(x1,y1,z1)andQ(x2,y2,z2)meettheplaneatR.LetRdividedPQintheratioλ:1.ThenthecoordinatesofRare
.Thispointliesontheplanegivenby(1).
Note12.8.1:If(ax1+by1+cz1+d)and(ax2+by2+cz2+d)areofthesamesignthenλisnegative.ThenthepointRdividesPQexternallyandsothepointsPandQlieonthesamesideoftheplane.
Note12.8.2:IfP(x1,y1,z1)andtheoriginlieonthesamesideoftheplaneax+by+cz+d=0ifax1+by1+cz1+danddofthesamesign.
12.10PLANEPASSINGTHROUGHTHEINTERSECTIONOFTWOGIVENPLANES
Letthetwogivenplanesbe
Thenconsidertheequation(ax1+by1+cz1+d1)+λ(ax2+by2+cz2+d2)=0.Thisbeingthefirstdegreeequationinx,yandz,representsaplane.Let(x1,y1,z1)bethepointonthelineoftheintersectionofplanesgivenbyequations(12.24)and(12.25).Then(x1,y1,z1)liesonthetwogivenplanes.
Then,clearly(a1x1+b1y1+c1z1+d1)+λ(a2x1+b2y1+c2z2+d2)=0.Fromthisequation,weinferthatthepoint(x1,y1,z1)liesontheplanegiven
by(12.26).Similarly,everypointinthelineofintersectionoftheplanes(12.24)and(12.25)lieontheplanes(12.24)and(12.25).
Hence,equation(12.26)istheplanepassingthroughtheintersectionofthetwogivenplanes.
12.11EQUATIONOFTHEPLANESWHICHBISECTTHEANGLEBETWEENTWOGIVENPLANES
Findtheequationoftheplaneswhichbisecttheanglebetweentwogivenplanes.Letthetwogivenplanesbe
LetP(x1,y1,z1)beapointoneitherofthebisectorsoftheanglebetweenthetwogivenplanes.ThentheperpendiculardistancefromPtothetwogivenplanesareequalinmagnitude.
Bytakingthepositivesign,wegettheequationofoneofthebisectorsandbytakingthenegativesign,wegettheequationtotheotherbisector.
Note12.11.1:Wecandeterminewhichofthetwoplanesbisectstheacuteanglebetweentheplanes.Forthis,wehavetofindtheangleθbetweenthebisectorplanesandoneofthetwogivenplanes.Iftanθ<1(θ<45°),thenthebisectorplanetakenistheinternalbisectorandtheotherbisectorplaneistheexternalbisector.Iftanθ>1thenthebisectorplanetakenistheexternalbisectorandtheotherbisectorplaneistheinternalbisector.
Note12.11.2:Wecanalsodeterminetheequationoftheplanebisectingtheanglebetweentheplanesthatcontaintheorigin.Supposetheequationofthetwoplanesarea1x+b1y+c1z+d1=0anda2x+b2y+c2z+d2=0whered1andd2
arepositive.LetP(x1,y1,z1)beapointonthebisectorbetweentheanglesoftheplanescontainingtheorigin.Thend1anda1x+b1y+c1z+dareofthesign.Sinced1ispositive,a1x+b1y+c1zisalsopositive.Similarly,a2x+b2y+c2z+d2isalsopositive.Therefore,theequationoftheplanebisectingtheangle
containingtheoriginis
Theequationofbisectorplanenotcontainingtheoriginis
12.12CONDITIONFORTHEHOMOGENOUSEQUATIONOFTHESECONDDEGREETOREPRESENTAPAIROFPLANES
Theequationthatrepresentsapairofplanesbeax2+by2+cz2+2fyz+2gzx+2hxy=0.Letthetwoplanesrepresentedbytheabovehomogenousequationofthe
seconddegreeinx,yandzbelx+my+nz=0andl1x+m1y+n1z=0.Then
Comparingtheliketermsonbothsides,weget
Thisistherequiredcondition.
Note12.12.1:Tofindtheanglebetweenthetwoplanes:
Letθbetheanglebetweenthetwoplanes.Then
Note12.12.2:Iftheplanesareperpendicularthenθ=90°andtheconditionforthatisa+b+c=0.
ILLUSTRATIVEEXAMPLES
Example12.1
Thefootoftheperpendicularfromtheorigintoaplaneis(13,–4,–3).Findtheequationoftheplane.
Solution
Thelinejoiningthepoints(0,0,0)and(13,–4,–3)isnormaltotheplane.Therefore,thedirectionratiosofthenormaltotheplaneare(13,–4,–3).Theequationoftheplaneisa(x–x1)+b(y–y1)+c(z–z1)=0
Example12.2
AplanemeetsthecoordinateaxesatA,BandCsuchthatthecentroidofthetriangleisthepoint(a,b,c).Findtheequationoftheplane.
Solution
Lettheequationoftheplanebe ThenthecoordinatesofA,BandC
are(α,0,0),(0,β,0)and(0,0,γ).ThecentroidofthetriangleABCis .
Butthecentroidisgivenas(a,b,c).
Therefore,theequationoftheplaneis
Example12.3
Findtheequationoftheplanepassingthroughthepoints(2,2,1),(2,3,2)and(–1,3,1).
Solution
Theequationoftheplanepassingthroughthepoint(2,2,1)isoftheforma(x–2)+b(y–2)+c(z–1)=0.Thisplanepassesthroughthepoints(2,3,2)and(–1,3,1).
∴0a+b+c=0and–3a+b+c=0
Solvingweget
Therefore,theequationoftheplaneis1(x–2)+3(y–2)–3(z–1)=0.
∴x+3y–3z–5=0
Example12.4
Findtheequationoftheplanepassingthroughthepoint(2,–3,4)andparalleltotheplane2x–5y–7z+15=0.
Solution
Theequationoftheplaneparallelto2x–5y–7z+15=0is2x–5y–7z+k=0.Thisplanepassesthroughthepoint(2,–3,4).
∴4+15–28+k=0ork=9Hence,theequationoftherequiredplaneis2x–5y–7z+9=0.
Example12.5
Findtheequationoftheplanepassingthroughthepoint(2,2,4)andperpendiculartotheplanes2x–2y–4z–3=0and3x+y+6z–4=0.
Solution
Anyplanepassingthrough(2,2,4)isa(x–2)+b(y–2)+c(z–4)=0.Thisplaneisperpendiculartotheplanes2x–2y–4z–3=0and3x+y+6z–
4=0.
Therefore,thedirectionratiosofthenormaltotherequiredplaneare1,3,–1.Therefore,theequationoftheplaneis(x–2)+0+(z–4)=0(i.e.)(x–2)+
3(y–2)–(z–4)=0.
x+3y–z–4=0
Example12.6
Determinetheconstantsksothattheplanesx–2y+kz=0and2x+5y–z=0areatrightanglesandinthatcasefindtheplanethroughthepoint(1,–1,–1)andperpendiculartoboththegivenplanes.
Solution
Theplanesx–2y+kz=0and2x+5y–z=0areperpendicular.Therefore,2–10–k=0∴k=–8.Anyplanepassingthrough(1,–1,–1)isa(x–1)+b(y+1)+C(x+1)=0.
Thisplaneisperpendiculartotheplanesx–2y–8z=0and2x+5y–z=0.
Therefore,theequationoftherequiredplaneis14(x–1)–5(y+1)+3(z+1)=0or14x–5y+3z–16=0.
Example12.7
AvariableplaneisataconstantdistancepfromtheoriginandmeetstheaxesinA,BandC.ShowthatthelocusofthecentroidofthetetrahedronOABCisx–2+y–2+z–2=16p–2.
Solution
LettheequationoftheplaneABCbe ThenthecoordinatesofO,A,B
andCare(0,0,0),(a,0,0),(0,b,0)and(0,0,c).LetthecentroidofthetetrahedronOABCbe(x1,y1,z1).Butthecentroidof
thetetrahedronis
TheperpendiculardistancefromOandtheplaneABCisp.
Thelocusof(x1,y1,z1)isx–2+y–2+z–2=16p–2.
Example12.8
Twosystemsofrectangularaxeshavethesameorigin.Ifaplanecutsthematdistances(a,b,c)and(a1,b1,c1)respectively,fromtheorigin,provethat
Solution
Let(o,x,y,z)and(O,X,Y,Z)bethetwosystemofcoordinateaxes.Theequationoftheplanewithrespectivefirstsystemofcoordinateaxisis
Theequationofthesameplanewithrespecttothesecondsystemofcoordinateaxesis
Theperpendiculardistancefromtheorigintotheplanegivenbytheequation(12.30)is
Theperpendiculardistancefromtheorigintotheplaneisgivenbytheequation
(12.31)is
Sincetheequations(12.30)and(12.31)representthesameplanethesetwoperpendiculardistancesareequal.
Example12.9
Avariableplanepassesthroughafixedpoint(a,b,c)andmeetsthecoordinateaxesinA,BandC.Provethatthelocusofthepointofintersectionofplanes
throughA,BandCparalleltothecoordinateplanesis
Solution
Lettheequationoftheplanebe
Thisplanepassesthroughthepoint(a,b,c).
ThentheequationoftheplanesthroughA,BandCparalleltothecoordinateplanesarex=α,y=βandz=γ.Let(x1,y1,z1)bethepointofintersectionoftheseplanes.Thenx1=α,y1=βandz1=γ
Therefore,fromequation(12.33),weget Thelocusof(x1,y1,z1)
is
Example12.10
Avariableplanemakesinterceptsonthecoordinateaxes,thesumofwhosesquaresisconstantandisequaltok2.Provethatthelocusofthefootoftheperpendicularfromtheorigintotheplaneis(x2+y2+z2)(x–2+y–2+z–2)=k2.
Solution
Lettheequationoftheplanebe
wherea,bandcaretheinterceptsonthecoordinateaxes.Giventhat
LetP(x1,y1,z1)bethefootoftheperpendicularfromOonthisplane.The
directionratiosofthenormalOPare Therefore,theequationofthe
normalOPareax=by=cz.Since(x1,y1,z1)liesonthenormal,ax1=by1+cz1=t(say).
From(12.34)and(12.35),weget
Thepoint(x1,y,z1)alsoliesontheplane.
Eliminatinga,b,cfrom(12.36)and(12.38)
Eliminatingtfrom(12.38)and(12.39)
Therefore,thelocusof(x1,y1,z1)is
Example12.11
FindtheequationoftheplanewhichcutsthecoordinateaxesatA,B,andCsuchthatthecentroidofΔABCisatthepoint(–1,–2,–4).
Solution
Lettheequationoftheplanebe ThenthecoordinatesofA,BandC
are(a,0,0),(0,b,0),(0,0,c).ThecentroidofΔABCis .Butthecentroid
isgivenas(–1,–2,–4).
∴a=–3,b=–6,c=–12
HencetheequationoftheplaneABCis
(i.e.)4x+2y+z+12=0
Example12.12
Findtheequationoftheplanepassingthroughthepoint(–1,3,2)andperpendiculartotheplanesx+2y+2z=5and3x+3y+2z=8.
Solution
Theequationoftheplanepassingthroughthepoint(–1,3,2)isA(x+1)+B(y–3)+C(z–2)=0.Thisplaneisperpendiculartotheplanesx+2y+2z=5and3x+3y+2z=8.Iftwoplanesareperpendicularthentheirnormalsareperpendicular.ThedirectionratiosofthenormaltotherequiredplaneareA,BandC.Thedirectionratiosofthenormalstothegivenplanesare1,2,2and3,3,2.
Therefore,thedirectionratiosofthenormaltotherequiredplaneare2,–4,3.Theequationoftherequiredplaneis2(x+1)–4(y–3)+3(z–2)=0(i.e.)2x–4y+3z+8=0.
Example12.13
Findtheequationoftheplanepassingthroughthepoints(9,3,6)and(2,2,1)andperpendiculartotheplane2x+6y+6z–9+0.
Solution
Anyplanepassingthroughthepoint(9,3,6)is
Theplanealsopassesthroughthepoint(2,2,1).
Theplane(12.40)isperpendiculartotheplane.
Therefore,theequationoftherequiredplaneis3(x–9)+4(y–3)–5(z–6)=0.
∴3x+4y–5z=9
Example12.14
Showthatthefollowingpoints(0,–1,0),(2,1,–1),(1,1,1)and(3,3,0)arecoplanar.
Solution
Theequationoftheplanepassingthroughthepoint(0,–1,0)isAx+B(y+1)+Cz=0.Thisplanepassesthroughthepoints(2,1,–1)and(1,1,1).
Therefore,theequationoftheplaneis4x–3(y+1)+2z=0.
∴4x–3y+2z–3=0.Substitutingx=3,y=3,z=0weget12–9–3=0whichistrue.Therefore,theplanepassesthroughthepoints(3,3,0)andhencethefour
givenpointsarecoplanar.
Example12.15
Findforwhatvaluesofλ,thepoints(0,–1,λ),(4,5,1),(3,9,4)and(–4,4,4)arecoplanar.
Solution
Theequationoftheplanepassingthroughthepoint(4,5,1)isA(x–4)+B(y–5)+C(z–1)=0.Thisplanepassesthroughthepoints(3,9,4)and(−4,4,4).
Therefore,theequationofplaneis5(x–4)–7(y–5)+11(z–1)=0.
Ifthisplanepassesthroughthepoint(0,–1,λ)then0+7+11λ+4=0.∴λ=–1
Example12.16
Avariableplanemovesinsuchawaythatthesumofthereciprocalsoftheinterceptsonthecoordinateaxesisaconstant.Showthattheplanepassesthroughafixedpoint.
Solution
Lettheequationoftheplanebe
Giventhatthesumofthereciprocalsoftheinterceptsisaconstant.
(12.44)–(12.45)gives
Thisplanespassesthroughthefixedpoint
Example12.17
ApointPmovesonfixedplane andtheplanethroughP
perpendiculartoOPmeetstheaxesinA,BandC.IftheplanesthroughA,BandCareparalleltothecoordinatesplanesmeetinapointthenshowthatthelocus
ofQis
Solution
Thegivenplaneis
LetPbethepoint(α,β,γ).TheplanepassesthroughP.
TheequationoftheplanenormaltoOPis
Theinterceptsmadebythisplaneonthecoordinateaxesare
Iftheseplanesmeetat(x1,y1,z1)then
Nowwehavetoeliminateα,β,γusing(12.47)and(12.49).From(12.49),
From(12.47)and(12.49),
Therefore,thelocusof(x1,y1,z1)is from(12.50)and
(12.51).
Example12.18
IffromthepointP(a,b,c)perpendicularsPL,PMbedrawntoYZ-andZX-planes,findtheequationoftheplaneOLM.
Solution
Pisthepoint(a,b,c).PLisdrawnperpendiculartoYZ-plane.Therefore,thecoordinatesofLare(0,b,0).PMisdrawnperpendiculartoZX-plane.Therefore,thecoordinatesofMare(0,0,c).WehavetofindtheequationoftheplaneOLM.Theequationoftheplanepassingthrough(0,0,0)isAx+By+Cz=0.Thisplanealsopassesthrough(0,b,c),(a,0,c).
TheequationoftheplaneOLMisbcx+cay–abz=0.
Example12.19
Showthat isthecircumcentreofthetriangleformedbythepoints(1,1,
0),(1,2,1)and(–2,2,–1).
Solution
A,BandCarethepoints(1,1,0),(1,2,1)and(–2,2,–1)andPisthepoint
ToprovethatPisthecircumcentreofthetriangleABC,wehaveto
showthat:
i. thepointsP,A,BandCarecoplanarandii. PA=PB=PC.
Theequationoftheplanethroughthepoint(1,1,0)isA(x–1)+b(y–1)+C(z–0)=0.Thisplanealsopassesthrough(1,2,1)and(–2,2,–1).
Therefore,theequationoftheplaneABCis–2(x–1)–3(y–1)+3z=0.
Substitutingthecoordinatesof weget–1+6–0–5=0whichis
true.Therefore,thepointsP,A,BandCarecoplanar.Now
Therefore,PisthecircumcentreofthetriangleABC.
Example12.20
Findtheratioinwhichthelinejoiningthepoints(2,–1,4)and(6,2,4)isdividedbytheplanex+2y+3z+5=0.
Solution
Lettheplanex+2y+3z+5=0dividethelinejoiningthepointsP(2,–1,4)andQ(6,2,4)intheratioλ:1.
Thenthepointofdivisionis
Thispointliesontheplanex+2y+3z+5=0.
Therefore,theplanedividesthelineexternallyintheratio17:27.
Example12.21
Aplanetrianglewhosesidesareoflengtha,b,andcisplacedsothatthemiddlepointsofthesidesareontheaxes.Ifα,βandγareinterceptsontheaxesthen
showthattheequationoftheplaneis where
Solution
Theequationoftheplaneis
LettheplanemeettheaxesatL,M,Nrespectively.L(α,0,0),M(0,β,0),N(0,0,γ)
(12.52)+(12.53)–(12.54)gives,
Therefore,theequationoftheplane whereα,β,γaregivenby
(12.55),(12.56)and(12.57).Let(x1,y1,z1),(x2,y2,z2)and(x3,y3,z3)betheverticesoftheΔABC.Then
Addingweget2(x1+x2+x3)=2αorx1+x2+x3=α
Similarly,
Therefore,theverticesofthetriangleare(–α,β,γ)(α,–β,γ)and(α,β,–γ).
Example12.22
Findtheanglebetweentheplanes2x–y+z=6,x+y+2z=3.
Solution
Thedirectionratiosofthenormaltotheplanesare2,–1,1and1,1,2.The
directioncosinesofthenormalare .Ifθistheangle
betweentheplanes,then
Example12.23
Provethattheplanex+2y+2z=0,2x+y–2z=0areatrightangles.
Solution
Thedirectionratiosofthenormalstotheplanesare1,2,2and2,1,–2.Ifthelinesaretobeperpendicularthena1a2+b1b2+c1c2=0.Hence,a1a2+b1b2+c1c2=2+2–4=0.Therefore,thenormalsareperpendicularandhencetheplanesare
perpendicular.
Example12.24
Findtheequationoftheplanecontainingthelineofintersectionoftheplanesx+y+z–6=0,2x+3y+4z+5=0andpassingthroughthepoint(1,1,1).
Solution
Theequationofanyplanecontainingthelineisx+y+z–6=λ(2x+3y+4z+5)=0.Ifthislinepassesthroughthepoint(1,1,1)then,1+1+1–6+λ(2+3+4+5)=0.
Therefore,theequationoftherequiredplaneis
Example12.25
Findtheequationoftheplanewhichpassesthroughtheintersectionoftheplanes2x+3y+10z–8=0,2x–3y+7z–2=0andisperpendiculartotheplane3x–2y+4z–5=0.
Solution
Theequationofanyplanepassingthroughtheintersectionoftheplanes2x+3y+10z–8=0and2x–3y+7z–2=0is2x+3y+10z–8+λ(2x–3y+7z–2)=0.Thedirectionratiosofthenormaltothisplaneare2+2λ,3–3λ,10+7λ.The
directionratiosoftheplane3x–2y+4z–5=0are3,–2,4.Sincethesetwoplanesareperpendicular,3(2+2λ)–2(3–3λ)+4(10+7λ)=0.
Therefore,therequiredplaneis2x+3y+10z–8–(2x–3y+7z–λ)=0.
Example12.26
Theplanex–2y+3z=0isrotatedthrougharightangleaboutitslineofintersectionwiththeplane2x+3y–4z+2=0.Findtheequationoftheplaneinitsnewposition.
Solution
Theplanex–2y+3z=0isrotatedaboutthelineofintersectionoftheplanes
Thenewpositionoftheplane(12.58)passesthroughthelineofintersectionofthetwogivenplanes.Therefore,itsequationis
Theplane(12.60)isperpendiculartotheplane(12.58).
Therefore,theequationoftheplane(12.58)initsnewpositionis
Example12.27
Thelinelx+my=0isrotatedaboutitslineofintersectionwiththeplanez=0throughanangleα.Provethattheequationoftheplaneis
Solution
Anyplanepassingthroughtheintersectionoflx+my=0andz=0is
Theplanelx+my+λz=0isrotatedthroughanangleαalongtheplane(12.61).
Therefore,theequationoftheplaneinitsnewpositionisgivenby
Example12.28
Findtheequationoftheplanepassingthroughthelineofintersectionoftheplanes2x−y+5z−3=0and4x+2y−z+7=0andparalleltoz-axis.
Solution
Theequationoftheplanepassingthroughthelineofintersectionofthegivenplanesis2x−y+5z−3+λ(4x+2y−z+7)=0.Iftheplaneisparalleltoz-axis,itsnormalisperpendiculartoz-axis.Thedirectionsofthenormaltotheplaneare2+4λ,−1+2λ,5−λ.Thedirectionratiosofthez-axisare0,0,1.
Hence,theequationoftherequiredplaneis(2x−y+5z−3)+5(4x+2y−z+7)=0.
Example12.29
Findthedistanceofthepoints(2,3,−5),(3,4,7)fromtheplanex+2y−2z=9andprovethatthesepointslieontheoppositesidesoftheplane.
Solution
LetthelinejoiningthepointsP(2,3,−5)andQ(3,4,7)bedividedbytheplaneintheratioλ:1.
Therefore,thepointsPandQlieontheoppositesideoftheplane.Theperpendiculardistancefrom(2,3,−5)totheplanex+2y−2z−9=0is
Theperpendiculardistancefrom(3,4,7)totheplaneis
Note12.29.1:Sincep1andp2areofoppositesignsthepointsareontheoppositesidesoftheplane.
Example12.30
Provethatthepoints(2,3,−5)and(3,4,7)lieontheoppositesidesoftheplanewhichmeetstheaxesinA,BandCsuchthatthecentroidofthetriangleA,BandCisthepoints(1,2,4).
Solution
Lettheequationoftheplanebe .ThenthecoordinatesofA,BandC
are(a,0,0),(0,b,0)and(0,0,c).Thecentroidis Butthecentroidis
givenas(1,2,4).
Therefore,theequationoftheplaneABCis
LetthelinejoiningthepointsP(2,3,−5)andQ(3,4,7)bedividedbytheplaneintheratioλ:1.
Then
Therefore,thepointslieontheoppositesidesoftheplane.
Example12.31
Findthedistancebetweentheparallelplanes2x−2y+z+3=0,4x−4y+2z+5=0.
Solution
Let(x1,y1,z1)beapointontheplane2x−2y+z+3=0
Thenthedistancebetweentheparallelplanesisequaltothedistancefrom(x1,y1,z1)totheotherplane.
Note12.31.1:Thedistancebetweentheparallelplanesax+by+cz+d=0and
ax+by+cz+d1=0is
Ondividingtheequation4x−4y+2z+5=0by2,weget
Distancebetweentheplanes=
Example12.32
Aplaneisdrawnthroughthelineofx+y=1,z=0tomakeanangle
withtheplanex+y+z=0.Provethattwosuchplanescanbedrawn.Findtheir
equation.Showthattheanglebetweentheplanesis
Solution
Theequationoftheplanethroughtheline
Thedirectionratiosofthisplaneis1,1,λ.Alsothedirectionratiosoftheplanex+y+z=0are1,1,1.Ifθistheanglebetweenthesetwoplanesthen
From(12.63),theequationsoftherequiredplanesarex+y+2z=1and5x+
5y+2z−5=0.Ifθistheanglebetweenthesetwoplanesthen
Example12.33
Findthebisectorsoftheanglesbetweentheplanes2x−y+2z+3=0,3x−2y+6z+8=0;alsofindoutwhichplanebisectstheacuteangle.
Solution
Thetwogivenplanesare
Theequationsofthebisectorsare
Letθbetheanglebetweentheplanes(12.64)and(12.66)then
Henceθ>45°.Theplane5x−y−4z−3=0bisectstheobtuseanglebetweentheplanes(12.64)and(12.65).Therefore,23x−13y+32z+45=0bisectstheacuteanglebetweenthe
planes(12.64)and(12.65).
Example12.34
Provethattheequation2x2−2y2+4z2+2yz+6zx+3xy=0representsapairof
planesandanglebetweenthemis
Solution
Herea=2,b=−2,c=4,f=1,g=3,
Now,abc+2fgh−af2−bg2−ch2=0
⇒–16+9−2+18−9=0Hence,thegivenequationrepresentsapairofplanes.Letθbetheangle
betweentheplanes.Then
Exercises
SectionA
1. IfPisthepoint(2,3,−1),findtheequationoftheplanepassingthroughPandperpendiculartoOP.
Ans.:2x+3y−z−14=0
2. Thefootoftheperpendicularfromtheorigintoaplaneis(12,−4,−3).Finditsequation.Ans.:12x−4y−3z+69=0
3. Findtheinterceptsmadebytheplane4x−3y+2z−7=0onthecoordinateaxes.
Ans.:
4. AplanemeetsthecoordinateaxesatA,BandCsuchthatthecentroidofthetriangleisthepoint
(a,b,c).Showthattheequationoftheplaneis
5. Findtheequationoftheplanethatpassesthroughthepoint(2,−3,1)andisperpendiculartothelinejoiningthepoints(3,4,−1)and(2,−1,5).
Ans.:x+5y−6z+19=0
6. OistheorginandAisthepoint(a,b,c).FindtheequationoftheplaneperpendiculartoA.
Ans.:ax+by+cz−(a2+b2+c2)=0
7. Findtheequationoftheplanepassingthroughthepoints:i. (8,−2,2),(2,1,−4),(2,4,−6)ii. (2,2,1),(2,3,2),(−1,3,0)iii. (2,3,4),(−3,5,1),(4,−1,2)
Ans.:(i)2x−2y−2z=14,(ii)2x+3y−3z−7=0,(iii)x+y−z−1=0
8. Showthatthepoints(0,−1,−1),(4,5,1),(3,9,4)and(−4,4,4)lieonaplane.9. Showthatthepoints(0,−1,0),(2,1,−1),(1,1,1)and(−3,3,0)arecoplanar.10. Findtheequationoftheplanethroughthethreepoints(2,3,4),(−3,5,1)and(4,−1,2).Alsofind
theangleswhichthenormaltotheplanemakeswiththeaxesofreference.
Ans.:
11. Findtheequationoftheplanewhichpassesthroughthepoint(2,−3,4)andisparalleltotheplane2x−5y−7z+15=0.
Ans.:2x−5y−7z+9=0
12. Findtheequationoftheplanethrough(1,3,2)andperpendiculartotheplanesx+2y+2z−5=0and3x+3y+3z−8=0.
Ans.:2x−4y+3z+8=0
13. Findtheequationoftheplanewhichpassesthroughthepoint(2,2,4)andperpendiculartotheplanes2x−2y−4z+3=0and3x+y+6z−4=0.
Ans.:x−3y−z−4=0
14. Findtheequationoftheplanewhichpassesthroughthepoints(9,3,6)and(2,2,1)andperpendiculartotheplane2x+4y+6z−9=0.
Ans.:3x+4y−5z−9=0
15. Findtheequationofthestraightlinepassingthroughthepoints(−1,1,1)and(1,−1,1)andperpendiculartotheplanex+2y+2z−5=0.
Ans.:2x+2y−3z+3=0
16. Findtheequationoftheplanewhichpassesthroughthepoints(2,3,1),(4,−5,3)andareparalleltothecoordinateaxes.
Ans.:y+4z−7=0,x−z−1=0,4x+y−11=0
17. Findtheequationoftheplanewhichpassesthepoint(1,2,3)andparallelto3x+4y−5z=0.Ans.:3x+4y−5z+4=0
18. Findtheequationoftheplanebisectingthelinejoiningthepoints(2,3,−1)and(−5,6,3)atrightangles.
Ans.:x−y−z+7=0
19. AvariableplaneisataconstantdistancepfromtheoriginandmeetstheaxesinA,BandC.ShowthatthelocusofcentroidofthetetrahedronOABCis
x−2+y−2+z−2=16p−2.20. OABCisatetrahedroninwhichOA,OBandOCaremutuallyperpendicular.Provethatthe
perpendicularfromOtothebaseABCmeetsitatitsorthocentre.21. ThroughthepointP(a,b,c)aplaneisdrawnatrightanglestoOPtomeettheaxesinA,BandC.
ProvethattheareaofthetriangleABCis wherepisthelengthofOP.
22. AplanecontainsthepointsA(−4,9,−9)andB(5,−9,6)andisperpendiculartothelinewhichjoinsBandC(4,−6,k).Obtainkandtheequationoftheplane.
Ans.:
23. Findthedistancebetweentheparallelplanes2x+y+2z−8=0and4x+2y+4z+5=0.
Ans.:
24. Findthelocusofthepoint,thesumofthesquaresofwhosedistancesfromtheplanesx+y+z=
0,x=z=0,x−2y+z=0is9.
Ans.:x2+y2+z2=9
25. Findtheequationoftheplanewhichisatadistance1unitfromtheoriginandparalleltotheplane3x+2y−z+2=0.
Ans.:
26. Theplane meetsthecoordinateaxesinA,BandC,respectively.
ShowthattheareaofthetriangleABCis
27. Showthattheequationsby+cz+d=0,cz+ax+d=0,ax+by+d=0representplanesparalleltoOX,OYandOZ,respectively.
28. Showthatthepoints(2,3,−5)and(3,4,7)lieontheoppositesidesoftheplanemeetingtheaxesinA,BandCsuchthatthecentroidofthetriangleABCisthepoint(1,2,4).
29. Findthelocusofthepointsuchthatthesumofthesquaresofitsdistancesfromtheplanesx+y+z=0andx−2y+z=0isequaltoitsdistancefromtheplanex−z=0.
Ans.:y2−2xz=0
30. Findthelocusofthepointwhosedistancefromtheoriginis7timesitsdistancefromtheplane2x+3y−6z=2.
Ans.:3x2+8y2+53z2−36yz−24zx+12xy−8x−12y+24z+14=0
31. Provethattheequationoftheplanepassingthroughthepoints(1,1,1),(1,−1,1)and(−7,−3,−5)andisparalleltoaxisofy.
32. Determinetheconstantksothattheplanesx−2y+kz=0and2x+5y−z=0areatrightangles,andinthatcasefindtheplanethroughthepoint(1,−1,−1)andperpendiculartoboththegivenplanes.
Ans.:k=−8,14x−5y+3z−16=0
33. Provethat3x–y–z+11=0istheequationoftheplanethrough(−1,6,2)andperpendiculartothejoinofthepoints(1,2,3)and(−2,3,4).
34. A,BandCarepoints(a,0,0),(0,b,0)and(0,0,c).FindtheequationoftheplanethroughBCwhichbisectsOA.BysymmetrywritedowntheequationsoftheplanethroughCAbisectingOB
andthroughABbisectingOC.Showthattheseplanespassthrough
SectionB
1. Findtheequationoftheplanethroughtheintersectionoftheplanesx+3y+6=0and3x−y−4z=0whoseperpendiculardistancefromtheoriginisunity.
Ans.:2x+y−2z+3=0,x−2y−2z−3=0
2. Findtheequationoftheplanethroughtheintersectionoftheplanesx−2y+3z+4=0and2x−3y+4z−7=0andthepoint(1,−1,1).
Ans.:9x−13y−17z−39=0
3. Findtheequationoftheplanethroughtheintersectionoftheplanesx+2y+3z+4=0and4x+3y+3z+1=0andperpendiculartotheplanex+y+z+9=0andshowthatitisperpendiculartoxz-plane.
Ans.:x−z=2
4. Findtheequationoftheplanethroughthepoint(1,−2,3)andtheintersectionoftheplanes2x−y+4z−7=0andx+2y−3z+8=0.
Ans.:17x+14y+11z+44=0
5. Findtheequationoftheplanepassingthroughtheintersectionoftheplanesx+2y+3z+4=0and4x+3y+2z+1=0andthroughthepoint(1,2,3).
Ans.:11x+4y−3z−10=0
6. Findtheequationoftheplanepassingthroughthelineofintersectionoftheplanesx−2y−z+3=0and3x+5y−2z−1=0whichisperpendiculartotheyz-plane.
Ans.:11y+z−10=0
7. Theplanex+4y−5z+2=0isrotatedthrougharightangleaboutitslineofintersectionwiththeplane3x+2y+z+1=0.Findtheequationoftheplaneinitsnewposition.
Ans.:20x+10y+12z+5=0
8. Aretheplanesgivenbytheequations3x+4y+5z+10=0and9x+12y+15z+20=0parallel?Ifsofindthedistancebetweenthem.
Ans.:
9. Findtheequationoftheplanepassingthroughthelineofintersectionoftheplanes2x−y=0and3x−y=0anperpendiculartotheplane4x+3y−3z=8.
Ans.:24x−17y+15z=0
10. Findtheequationoftheplanepassingthroughthelineofintersectionoftheplanesax+by+cz+d=0anda1x+b1y+c1z+d1=0perpendiculartoxy-plane.
Ans.:(ac1−a1c)x+(bc1−b1c)y+(dc1−d1c)z=0
11. Findtheequationoftheplanepassingthroughthelineofintersectionoftheplanes2x+3y+10z−8=0,2x−3y+7z−2=0andisperpendiculartotheplane3x−2y+4z−5=0.
Ans.:2y+z−2=0
12. Obtaintheequationoftheplanesbisectingtheanglesbetweentheplanesx+2y−2z+1=0and12x−4y+3z+5=0.Alsoshowthatthesetwoplanesareatrightangles.
Ans.:23x−38y+35z+2=049x+14y−17z+28=0
13. Findtheequationoftheplanethatbisectstheanglebetweentheplanes3x−6y−2z+5=0and4x−12y+3z−3=0whichcontaintheorigin.Doesthisplanebisecttheacuteangle?
Ans.:yes,67x+162y+47z+44=0
14. Findtheequationoftheplanethatbisectstheacuteanglebetweentheplanes3x−4y+12z−26=0andx+2y−2z−9=0.
Ans.:22x+14y+10z−195=0
15. Findtheequationoftheplanethatbisectstheobtuseanglebetweentheplanes4x+3y−5z+1=0and12x+5y−13=0.
Ans.:8x−14y−13=0
16. Showthattheoriginliesintheacuteanglebetweentheplanesx+2y−2z−9=0,4x−3y+12z+13=0.Findtheplanesbisectingtheanglebetweenthemandfindtheplanewhichbisectstheacuteangle.
Ans.:x+35y−10z−156=0
17. Findtheequationoftheplanewhichbisectstheacuteanglebetweentheplanesx+2y+2z−3=0and3x+4y+12z+1=0.
Ans.:11x+19y+31z−18=0
18. Provethattheequation2x2−6y2–12z2+18yz+2zx+xy=0representsapairofplanesand
showthattheanglebetweenthemis
19. Provethattheequation representsapairofplanes.
20. Iftheequationɸ(x,y,z)=ax2+by2+cz2+2fyz+2gzx+2hxy=0representsapairofplanes
thenprovethattheproductofthedistancesoftheplanesfrom
Chapter13
StraightLine
13.1INTRODUCTION
TheintersectionoftwoplanesP1andP2isthelocusofallthecommonpointsonboththeplanesP1andP2.Thislocusisastraightline.Anygivenlinecanbeuniquelydeterminedbyanyofthetwoplanescontainingtheline.Thus,alinecanberegardedasthelocusofthecommonpointsoftwointersectingplanes.Letusconsiderthetwoplanes
Anysetofcoordinates(x,y,z)whichsatisfythesetwoequationssimultaneouslywillrepresentapointonthelineofintersectionofthesetwoplanes.Hencethesetwoequationstakentogetherwillrepresentastraightline.Itcanbenotedthattheequationofx–axisarey=0,z=0.Theequationofthey–axisisx=0,z=0andtheequationofthez–axisisx=0,y=0.Therepresentationofthestraightlinebytheequationsax+by+cz+d=0anda1x+b1y+c1z+d1=0iscallednon–symmetricalform.Letusnowderivetheequationsofastraightlineinthesymmetricalform.
13.2EQUATIONOFASTRAIGHTLINEINSYMMETRICALFORM
LetA(x1,y1,z1)beapointonthestraightlineandP(x,y,z)beanypointonthestraightline.Letl,m,nbethedirectioncosinesofthestraightline.LetOP=r.TheprojectionsofAPonthecoordinateaxesarex−x1,y−y1,z−z1.AlsotheprojectionsofAPonthecoordinateaxesaregivenbylr,mr,nr.Thenx−x1=lr,y−y1=mrandz−z1=nr.
Theseequationsarecalledthesymmetricalformofthestraightline.
Aliter:Wenowderivetheequationsinsymmetricalformfromthevectorequationofthestraightlinepassingthroughapointandparalleltoavector.
LetAbeagivenpointonastraightlineandPbeanypointonthestraight
line.Let beavectorparalleltotheline.LetObetheoriginand
Then
But
wheretisascalar.From(13.3)and(13.4),
ThisequationistrueforallpositionsofPonthestraightlineandthereforethisisthevectorequationofthestraightline.Let
Thenfromequation(13.4),wehave
Equatingthecoefficientsof , and ,wehave
Thesearethecartesianequationsofthestraightlineinsymmetricalform.
Note13.2.1:Toexpresstheequationsofastraightlineinsymmetricalformwerequire(i)thecoordinateofapointonthelineand(ii)thedirectioncosinesofthestraightline.
Note13.2.2:Anypointonthislineis(x1+lr,y1+mr,z1+nr).Evenifl,mandnarethedirectionratiosoftheline,(x1+lr,y1+mr,z1+nr)willrepresentapointonthelinebutrwillnotbedistancebetweenthepoints(x,y,z)and(x1,y1,z1).
13.3EQUATIONSOFASTRAIGHTLINEPASSINGTHROUGHTHETWOGIVENPOINTS
LetP(x1,y1,z1)andQ(x2,y2,z2)betwogivenpoints.Thedirectionratiosofthelinearex2−x1,y2−y1,z2−z1.
Therefore,theequationsofthestraightlineare .
Aliter:LetObetheoriginandPandQbethepointsonthestraightlineandRbeanypointonthestraightline.
Then
But
Thisisthevectorequationofthestraightline.
Let
Thenfrom(13.6),weget
Equatingthecoefficientsof , and ,weget
13.4EQUATIONSOFASTRAIGHTLINEDETERMINEDBYAPAIROFPLANESINSYMMETRICALFORM
Wehavealreadyseenthatastraightlineisdeterminedbyapairofplanesax+by+cz+d=0anda1x+b1y+c1z+d1=0wenowexpresstheseequationsinsymmetricalform.Tofinditweneedtofindthedirectioncosinesofthelineandthecoordinates
ofapointontheline.Letl,m,nbethedirectioncosinesoftheline.Thislineisperpendiculartothenormaltothetwogivenplanessincethelineliesontheplane.Thedirectionratiosofthetwonormalsarea1,b1,c1anda2,b2,c2.Thedirectioncosinesofthelinearel,m,n.Sincethenormalsareperpendiculartothelinewehave,
Solvingforl,m,nweget
Therefore,thedirectionratiosofthelineare
Tofindapointontheline,letusfindthepointwherethelinemeetstheplane.z=0anda1x+b1y+d=0anda1x+b1y+d1=0.Solvingthelasttwoequations,weget
Therefore,apointonthelineis
Thentheequationsofthestraightlinesare .
Note13.4.1:Wecanalsofindthepointwherethelinemeetstheyz–planeorzx–plane.
13.5ANGLEBETWEENAPLANEANDALINE
Lettheequationoftheplanebeax+by+cz+d=0.
Lettheequationofthelinebe .
Letθbetheanglebetweentheplaneandtheline.Thedirectionratiosofthenormaltotheplanearea,b,c.Thedirectionratiosofthelinearel,m,n.Sinceθ
istheanglebetweentheplaneandtheline, istheanglebetweenthe
normaltotheplaneandtheline.
Note13.5.1:Ifthelineisparalleltotheplane,θ=0.
∴al+bm+cn=0
13.6CONDITIONFORALINETOBEPARALLELTOAPLANE
Lettheequationoftheplanebe
Lettheequationofthelinebe
Ifthelineisparalleltotheplanethenthenormaltotheplaneisperpendiculartotheline.Theconditionforthisis
Since(x1,y1,z1)isapointonthelineanddoesnotlieontheplanegivenby(13.10).
∴ax1+by1+cz1+d≠0Hencetheconditionsfortheline(13.11)tobeparalleltotheplane(13.10)are
al+bm+cn=0andax1+by1+cz1+d≠0.
13.7CONDITIONSFORALINETOLIEONAPLANE
Lettheequationofthelinebe
Lettheequationoftheplanebe
Sincethelineliesontheplane,
Sincethelineliesontheplaneeverypointonthelineisalsoapointontheplane.(x1,y1,z1)isapointonthelineandthereforeitshouldalsolieontheplanegivenby(13.14).Hence,ax1+by1+cz1+d=0.Therefore,theconditionsfortheline(13.13)tobeparalleltotheplane(13.14)areal+bm+cn=0andax1+by1+cz1+d=0.
13.8TOFINDTHELENGTHOFTHEPERPENDICULARFROMAGIVENPOINTONALINE
LetthegivenpointbeP(p,q,r)andthegivenlineQRbe
ThenL(x1,y1,z1)isapointontheline.DrawPMperpendiculartotheline.
AlsoLMistheprojectionofPLonQR.
Thenfrom(13.17),
13.9COPLANARLINES
Findtheconditionforthelines and
tobecoplanarandalsofindtheequationoftheplane
containingthesetwolines.Considerequations,
Lettheequationoftheplanebe
Sincetheplanescontainslines(13.20),wehave
From(13.20)and(13.21),weget
Sincetheplanealsocontainstheline(13.21)thepoint(x2,y2,z2)liesontheplane(13.22).
Eliminatinga,b,cfromequation(13.24),(13.26)and(13.27),weget
Thisistherequiredconditionforthelines(13.20)and(13.21)tobecoplanar.Eliminatinga,b,cfromtheequation(13.23),(13.24)and(13.25),wegetthe
equationoftheplanecontainingthetwogivenlinesas
Aliter:Iftheplanesarecoplanartheymayintersect.Anypointontheline(13.20)isx1+l1r1,y1+m1r1,z1+n1r1.Anypointontheline(13.21)isx2+l2r2,y2+m2r2,z2+n2r2.Ifthetwolinesintersectthenthetwopointsarethesame.
Eliminatingr1andr2fromtheaboveequations,weget
Thisistherequiredconditionforcoplanarlines.
13.10SKEWLINES
Twonon–intersectingandnon–parallellinesarecalledskewlines.Therealsoexistsashortestdistancebetweentheskewlinesandthelineoftheshortestdistancewhichiscommonperpendiculartobothofthese.
13.10.1LengthandEquationsoftheLineoftheShortestDistance
Lettheequationoftheskewlinesbe
LetPQbethelineoftheshortestdistancebetweenlines(13.28)and(13.29).Letl,m,nbethedirectioncosinesofthelinesoftheshortestdistancePQ.
TheconditionforPQtobeperpendiculartoABandCDare
Solvingthesetwo,weget
Therefore,thedirectionratiosofthelinePQarem1n2−m2n1,n1l2−n2l1,l1m2−l2m1.Therefore,thedirectionratiosofthelineoftheSDare
l(x1,y1,z1)andm(x2,y2,z2)arepointsonthelines(13.28)and(13.29).ThenthelengthoftheSD=PQ=ProjectionofLMonPQ=(x1−x2)l+(y1−
y2)m+(z1−z2)n,wherel,m,narethedirectioncosinesofthelinePQ.
TheequationoftheplanecontainingthelinesABandPQis
TheequationoftheplanecontainingthelinesCDandPQis
Therefore,theequationofthelineoftheSDistheintersectionofthesetwoplanesanditsequationsaregivenby
Note13.10.1:Ifthelines(13.28)and(13.29)arecoplanarthentheSDbetweenthelinesiszero.Hencetheconditionforthelines(13.28)and(13.29)tobe
coplanar,from(13.30)is
Aliter:Letthevectorequationsofthetwolinesbe
wheretandsarescalars.
Ifthelines(13.33)and(13.34)arecoplanarthentheplaneisparalleltothe
vectors and .Thereby isperpendiculartotheplanecontaining and .
Alsoas and arethepointsontheplane, isalineontheplaneandis
perpendicularto .Theconditionforthisis or
Thescalarformoftheequationis .Thevector
equationoftheplanecontainingthetwolinesis or
But .
Therefore,thescalarequationoftheplaneis
Aliter:
Letthevectorequationofthetwolinesbe and
Let and , .
LetDQbetheSDbetweenthelinesABatCD.Then isperpendicularto
both and .Then isparallelto .Let and bethepositionvectorsof
pointsLandMonABandCD,respectively.
13.10.2EquationoftheLineofSD
TheequationofthelineoftheshortestdistanceistheequationofthelineofintersectionoftheplanesthroughthegivenlinesandtheSD.Theequationofthe
planecontainingtheline andtheSDPQisparallelto and
thereforeperpendicularto is
Similarlytheequationoftheplanecontainingtheline andPQis
TheequationofthelineofSDistheequationofthelineofintersectionof(13.35)and(13.36).(i.e.)
Inscalarforms,theequationofthelineare
13.11EQUATIONSOFTWONON-INTERSECTINGLINES
Wewillnowshowthattheequationsofanytwoskewlinescanbepartintotheformy=mx,z=candy=−mx,z=−c.
LetABandCDbetwoskewlines.LetLMbethecommonperpendiculartotheskewlines.LetLM=2candθbe
itsmiddlepoint.ChooseObetheoriginanddrawlinesOPandOQparallelto
ABandCD,respectively.Letthebisectorsof bechosenasaxesofxandy.
LetOEbetakenasz–axis.Let sothat .ThenthelineOP
makesangleθ, and withx−,y−,z−axes.Itsdirectioncosinesarecosα,
sinα,0.ThecoordinatesofLare(0,0,c).ABisastraightlinepassingthroughL
andparalleltoAB.TheequationsofthelineOPare ory=
xtanθ,z=c(i.e.)y=mx,z=cwherem=tanθ
ThelineOQmakesangles−θ, −θand withx-,y-,z-axes.
ThedirectioncosinesofthelineOQarecosθ,−sinθ,θ.ThecoordinatesofMare(0,0,−c).CDisastraightlinepassingthroughF
andparalleltoCD.Itsequationsare .
(i.e.)y=−mx,z=−cwherem=tanθ
Note13.11.1:AnypointonthelineABis(r1,mr1,c)andonaxis(r1,−mr1,−c).
13.12INTERSECTIONOFTHREEPLANES
Threeplanesmayintersectinalineorapoint.Letusfindtheconditionsforthreegivenplanestointersect(i)inalineand(ii)inapoint.
Lettheequationsofthreegivenplanesbe
Theequationofanyplanepassingthroughtheintersectionofplanes(13.37)and(13.38)is
Ifplanes(13.37),(13.38)and(13.39)intersectinalinethenequations(13.39)and(13.40)representthesameplaneforsamevaluesofλ.Identifyingequations(13.40)and(13.39),weget
Eliminatingλandμfromtheequationtakenthreeatatime,weget
Therefore,theconditionsforthethreeplanestointersectinalineareΔ1=0,Δ2=0,Δ3=0andΔ4=0.
Note13.12.1:Ofthesefourconditionsonlytwoareindependentsinceiftwoplaneshavetwopointsincommonthentheyshowthelinejoiningthesetwopointsshouldalsohaveincommon.Itcanbeprovedifanytwooftheseconditionsaresatisfied,thentheothertwowillalsosatisfy.
Aliter:Theequationsofthelineofintersectionof(13.37)and(13.38)aregivenby
Iftheplanes(13.37),(13.38)and(13.39)intersectinaplanethentheconditionsare(i)theline(13.41)mustbeparalleltotheplane(13.39)and(ii)thepoint
mustlieontheplanegivenby(13.39).Theconditionsfor
(13.37)is
Thecondition(ii)isgivenby
Therefore,theconditionsforplanestointersectinalineareΔ3=0andΔ4=0(ii)Conditionfortheplanetointersectatapoint:Solvingequations(13.37),(13.38)and(13.39),weget
IftheplanesintersectatapointthenΔ4≠0.HencetheconditionforaplanetointersectatapointisΔ4≠0.
Aliter:Iftheplanesmeetatapointthenthelineofintersectionofanytwoplanesisnon–paralleltothethirdplane.Letl,m,nbethedirectioncosinesoftheintersectionofplanes(13.37)and(13.38).Then
Solvingthetwoequationsforl,m,nweget,
Therefore,thedirectionratiosofthelinesareb1c2−b2c1,a2c1−a1c2,a1b2−a2b1.Alsothelineofintersectionwillnotbeparalleltothethirdplane.
Thisistherequiredcondition.
13.13CONDITIONSFORTHREEGIVENPLANESTOFORMATRIANGULARPRISM
Thelineofintersectionofplanes(13.37)and(13.38)isgivenby
Thethreeplanesformatriangularprismifthelineisparalleltothethirdplane.Theconditionsforthisarethelineisnormaltotheplane(13.39)andthe
point doesnotlieontheplane(13.39).
(i.e.)Δ4=0andΔ3≠0.Thesearetherequiredconditions.
ILLUSTRATIVEEXAMPLES
Example13.1
Findtheequationofthelinejoiningthepoints(2,3,5)and(−1,2,−4).
Solution
Thedirectionratiosofthelineare2+1,3−2,5+4(i.e.)3,1,9.Therefore,the
equationsofthelineare .
Example13.2
Findtheequationofthelinepassingthroughthepoint(3,2,−6)andperpendiculartotheplane3x−y−2z+2=0.
Solution
Thedirectionratiosofthelinearethedirectionratiosofthenormaltotheplane.Therefore,thedirectionratiosofthelineare3,−1,−2.Giventhat(4,2,−6)isapointontheplane.
Therefore,theequationsofthelineare .
Example13.3
Findtheequationsofthelinepassingthroughthepoint(1,2,3)andperpendiculartotheplanesx−2y−z+5=0andx+y+3z+6=0.
Solution
Letl,m,nbethedirectionratiosofthelineofintersectionoftheplanesx−2y−z+5=0andx+y+3z+6=0.Thenl−2m−n=0andl+m+3n=0
Sincethelinealsopassesthroughthepoint(−1,2,3),itsequationsis
Example13.4
Expressthesymmetricalformoftheequationsofthelinex+2y+z−3=0,6x+8y+3z−13=0.
Solution
Toexpresstheequationsofalineinsymmetricalformwehavetofind(i)thedirectionratiosofthelineand(ii)apointontheline.Letl,m,nbethedirectionratiosofline.Thenl+2m+n=0and6l+8m+3n
=0.
Letusfindthepointwherethelinemeetsthexy–plane(i.e.)z=0.
Therefore,theequationsofthelineare
Example13.5
Findtheperpendiculardistancefromthepoint(1,3,−1)totheline
Solution
Theequationsofthelineare
Anypointonthislineare(5r+13,−8r−8,r+31).DrawPQperpendiculartotheplane.Thedirectionratiosofthelineare(5r+12,−8r−11,r+32).SincethelinePQisperpendiculartoQR,wehave
Qisthepoint(3,8,29)andPis(1,3,−1)
Example13.6
Findtheequationofplanepassingthroughtheline andparallel
totheline .
Solution
Anyplanecontainingtheline
is
where
Alsothelineisparalleltotheplane
SolvingforA,BandCfrom(13.44)and(13.46),weget or
Therefore,theequationoftherequiredplaneis11(x−1)+2(y+1)−5(z−3)=0.
(i.e.)11x+2y−5z+6=0
Example13.7
Findtheimageofthepoint(2,3,5)ontheplane2x+y−z+2=0.
Solution
LetQbetheimageofthepointP(2,3,5)ontheplane2x+y−z+2=0.The
equationofthelinePQis
Anypointonthislineis(2r+2,r+3,−r+5).Whenthelinemeetstheplane,thispointliesontheplane2x+y−z+2=0.
Example13.8
Findtheimageoftheline intheplane2x−y+z+3=0.
Solution
Theequationsofthelineare
Anypointonthislineis(3r+1,5r+3,2r+4).Asthispointliesontheplane,
HencethecoordinatesofRare(−5,−7,0).TheequationsofthelinePLperpendiculartotheplaneare
Anypointonthislineis(2r1+1,−r1+3,r1+4).Ifthispointliesontheplane(13.48),weget2(2r1+1)−(−r1+3)+(r1+4)+3=0.
(i.e.)6r1+6=0orr1=−1.Therefore,thecoordinatesofLwherethislinemeetstheplane(13.47)are(−1,4,3).IfQ(x1,y1,z1)istheimageofPintheplane
HencetheequationsofthereflectionlineRQare .
Example13.9
Findtheequationofthestraightlinesthroughtheorigineachofwhichintersects
thestraightline andareinclinedatanangleof60°toit.
Solution
TheequationsofthelinePQare
ThepointPonthislineisP(2r+3,r+3,r).
ThedirectionratiosofOPare2r+3,r+3,r.Since ,
orr2+3r+2=0orr=−1,−2.Therefore,thecoordinatesofPandQare(1,−2,−1)and(−1,1,−2).
HencetheequationsofthelinesOPandOQare and .
Example13.10
Findthecoordinatesofthepointwherethelinegivenbyx+3y−z=6,y−z=4cutstheplane2x+2y+z=6.
Solution
Letl,m,nbethedirectioncosinesofthelinex+3y−z=6,y−z=4.Then
Therefore,thedirectionratiosofthelineare2,−1,−1.Whentheline
meetsthexy–planewhoseequationisz=0,wehavex+3y=6,y=4.Therefore,thepointwherethelinemeetsxy–planeis(−6,4,0).
Therefore,theequationsofthelineare
Anypointonthislineis(2r−6,−r+4,−r).Thispointliesontheplane2x+2y+z=0.
Hencetherequiredpointis(2,0,−4).
Example13.11
Findthedistanceofthepoint(1,−2,3)fromtheplanex−y+z=5measured
paralleltotheline
Solution
Theequationsofthelinethrough(1,−2,3)andparalleltotheline are
.Anypointonthislineis(2r+1,3r−2,−6r+3).Ifthispoint
liesontheplanex−y+z=5then(2r+1)−(3r−2)+(−6r+3)=5.
(i.e.)−7r+1=0or
Therefore,thepointPis .
Therefore,thedistancebetweenthepointsA(1,−2,3)and is
.
Example13.12
Provethattheequationofthelinethroughthepoints(a,b,c)and(a′,b′,c′)passesthroughtheoriginifaa′+bb′+cc′=pp′wherepandp′arethedistancesofthepointsfromtheorigin.
Solution
Theequationsofthelinethrough(a,b,c)and(a′,b′,c′)are
Ifthispassesthroughtheoriginthen
Letpandp′bethedistancesofthepoints(a,b,c)and(a′,b′,c′)fromtheorigin.
ByLagrange’sidentity,
Example13.13
IffromthepointP(x,y,z),PMisdrawnperpendiculartotheline andis
producedtoQsuchthatPM=MQthenshowthat
Solution
TheequationofthelineOAis .
Anypointonthislineis(lr,mr,nr).
IfMisthispointthen
ThedirectionratiosofthelineMParex−lr,y−mr,z−nr.SinceMPisperpendiculartoOA,
From(13.50),weget .
Example13.14
Reducetheequationsofthelinesx=ay+b,z=cy+dtosymmetricalformandhencefindtheconditionthatthelinebeperpendiculartothelinewhoseequationsarex=a′y+b′,z=c′y+d′.
Solution
Theline
canbeexpressedinthesymmetricalformas
Theline
insymmetricalformis
Ifthelines(13.51)and(13.52)areperpendicularthenaa′+bb′+cc′=0.Thisistherequiredcondition.
Example13.15
FindtheequationofthelinepassingthroughGperpendiculartotheplaneXYZrepresentedbytheequationlx+my+nz=pwherel2+m2+n2=1andcalculatethedistanceofGfromtheplane.
Solution
TheequationoftheplaneXYZis
wherel2+m2+n2=1.Whenthisplanemeetsthex–axis,y=0andz=0.
HenceXisthepoint .Similarly,Yis andZis .
ThecentroidofΔXYZis .
TheequationofthelinethroughGperpendiculartotheplaneXYZis
WhenthislinemeetstheYOZplane,x=0(13.56)
Then
Here,risthedistanceofGfromtheplane(13.55)sincepandl2arepositive,r=GA
Example13.16
Findtheperpendiculardistanceofangularpointsofacubefromadiagonalwhichdoesnotpassthroughtheangularpoint.
Solution
Letabethesideofthecube.BB′isadiagonalofthecubenotpassingthroughO.ThedirectionratiosofBB′area,−a,a.(i.e.)1,−1,1.Thedirectioncosinesof
BB′are .TheprojectionsofOB′onBB′
Example13.17
Provethattheequationsofthelinethroughthepoint(α,β,γ)andatrightangles
tothelines are .
Solution
Letl,m,nbethedirectioncosinesofthelineperpendiculartothetwogivenlines.Thenwehave
Therefore,thedirectionratiosofthelinearem1n2−m2n1,n1l2−n2l1,l1m2−l2m1.Thelinealsopassesthroughthepoint(x1,y1,z1).
Itsequationsare
Exercises1
1. Showthattheline isparalleltotheplane2x+3y−z+4=0.
2. Findtheequationoftheplanethroughtheline andthepoint(0,7,−7).Show
furthertheplanecontainstheline .
Ans.:x+y+z=0
3. Findtheequationoftheplanewhichpassesthroughtheline3x+5y+7z−5=0=x+y+z−3andparalleltotheline4x+y+z=0=2x−3y−5z.
Ans.:2x+4y+y+6z=2
4. Findtheequationsofthelinethroughthepoint(1,0,7)whichintersecteachofthelines
Ans.:7x−6y−z=0,9x−7y−z−2=0
5. Findtheequationoftheplanewhichpassesthroughthepoint(5,1,2)andisperpendiculartothe
line Findalsothecoordinatesofthepointinwhichthislinecutstheplane.
Ans.:x−2y−2z−1=0;(1,2,3)
6. Findtheequationoftheplanethrough(1,1,2)and(2,10,−1)andperpendiculartothestraight
line
Ans.:3x−y−7z+2=0
7. Findtheprojectionoftheline3x−y+2z=1,x+2y−z=2ontheplane3x+2y+z=0.Ans.:3x+2y+z=0,3x−8y+7z+4=0
8. Findtheprojectionofthelinex=3−6t,y=2t,z=3+2tintheplane3x+4y−5z−26=0.
Ans.:
9. Findtheequationoftheplanewhichcontainsthelineandisperpendiculartotheplanex+2y+z=12.
Ans.:9x−2y−5z+4=0
10. Findtheequationoftheplanewhichpassesthroughthez–axisandisperpendiculartotheline
.
Ans.:xcosα+ysinα=0
11. Findtheequationsoftwoplanesthroughtheoriginwhichareparalleltotheline
anddistant fromit.Showalsothatthetwoplanesareperpendicular.
Ans.:x+2y−2z=0,2x+2y+z=0
12. Findtheequationstothelineofthegreatestslopethroughthepoint(1,2,−1)intheplanex−2y+3z=0assumingthattheaxesaresoplacedthattheplane2x+3y−4z=0ishorizontal.
Ans.:
13. Assumingtheline asvertical,findtheequationofthelineofthegreatestslopeinthe
plane2x+y−5z=12andpassingthroughthepoint(2,3,−1).
Ans.:
14. Withthegivenaxesrectangulartheline isvertical.Findthedirectioncosinesofthe
lineofthegreatestslopeintheplane3x−2y+z=0andtheangleofthislinemakeswiththehorizontalplane.
Ans.:
15. Showthatthelines willbecoplanarif
16. Showthattheequationoftheplanethroughtheline andwhichisperpendiculartothe
planecontainingthelines and is∑(m−n)x=0.
17. Showthattheline and willlieinaplaneifα=βorβ=γ
orγ=α.
18. Findtheequationoftheplanepassingthroughtheline andperpendiculartothe
planex+2y+z=12.Ans.:9x−2y+5z+4=0
19. Findtheequationsofthelinethrough(3,4,0)andperpendiculartotheplane2x+4y+7z=8.
Ans.:
20. Findtheequationoftheplanepassingthroughtheline areparalleltotheline
.
Ans.:4y−3z+1=0,2x−7z+1=0,3x−2y+1=0.
21. Showthattheequationoftheplanesthroughthelinewhichbisecttheanglebetweenthelines
(wherel,m,nandl′,m′,n′aredirectioncosines)andperpendiculartothe
planecontainingthemare(l+l′)x+(m+m′)y+(n+n′)z=0.
22. Findtheequationoftheplanethroughtheline andparalleltothecoordinate
planes.Ans.:xcosθ+ysinθ=0
23. Provethattheplanethroughthepoint(α,β,γ)andthelinex=py+q=zx+risgivenby
.
24. ThelineLisgivenby .FindthedirectioncosinesoftheprojectionsofLonthe
plane2x+y−3z=4andtheequationoftheplanethroughLparalleltotheline2x+5y+3z=4,x−y−5z=6.
Ans.:
Exercises2
1. Findtheequationofthelinejoiningthepointsi. (2,3,5)and(−1,2,−4)ii. (1,−1,3)and(3,3,1)
Ans.:
2. Findtheequationsofthelinepassingthroughthepoint(3,2,−8)andisperpendiculartotheplane3x−y−2z+2=0.
Ans.:
3. Findtheequationsofthelinepassingthroughthepoint(3,1,−6)andparalleltoeachoftheplanesx+y+2z−4=0and2x−3y+z+5=0.
Ans.:
4. Findtheequationsofthelinethroughthepoint(1,2,3)andparalleltothelineofintersectionoftheplanesx−2y−z+5=0,x+y+3z−6=0.
Ans.:
5. Findthepointatwhichtheline meetstheplane2x+4y−z+1=0.
Ans.:
6. Findthecoordinatesofthepointatwhichtheline meetstheplane2x+3y+z=0.
Ans.:
7. Provethattheequationsofthenormaltotheplaneax+by+cz+d=0throughthepoint(α,β,γ)
are
8. Expressinsymmetricalformthefollowinglines:i. x+2y+z=3,6x+8y+3z=13ii. x−2y+3z−4=0,2x−3y+4z−5=0iii. x+3y−z−15=0,5x−2y+4z+8=0
Ans.:
9. Provethatthelines3x+2y+z−5=0,x+y−2z−3=0and8x−4y−4z=0,7x+10y−8z=0areatrightangles.
10. Provethatthelinesx−4y+2z=0,4x−y−3z=0andx+3y−5z+9=0,7x−5y−z+7=0areparallel.
11. Findthepointatwhichtheperpendicularfromtheoriginonthelinejoiningthepoints(−9,4,5)and(11,0,−1)meetsit.
Ans.:(1,2,2).
12. Provethatthelines2x+3y−4z=0,3x−4y+7=0and5x−y−3z+12=0,x−7y+5z−6=0areparallel.
13. Findtheperpendicularfromthepoint(1,3,9)totheline
Ans.:21
14. Findthedistanceofthepoint(−1,−5,−10)fromthepointofintersectionoftheline
andtheplanex−y+z=5.
Ans.:13
15. Findthelengthoftheperpendicularfromthepoint(5,4,−1)totheline .
Ans.:
16. Findthefootoftheperpendicularfromthepoint(−1,11,5)totheline
Ans.:
17. Obtainthecoordinatesofthefootoftheperpendicularfromtheoriginonthelinejoiningthepoints(−9,4,5)and(11,0,−1).
18. Findtheimageofthepoint(4,5,−2)intheplanex−y+3z−4=0.Ans.:(6,3,4)
19. Findtheimageofthepoint(1,3,4)intheplane2x−y+z+3=0.Ans.:(1,0,7)
20. Findtheimageofthepoint(2,3,5)intheplane2x+y−z+2=0.
Ans.:
21. Findtheimageofthepoint(p,q,r)intheplane2x+y+z=6andhencefindtheimageoftheline
.
Ans.:
22. Findthecoordinatesofthefootoftheperpendicularfrom(1,0,2)totheline
Alsofindthelengthoftheperpendicular.
Ans.:
23. Findtheequationinsymmetricalformoftheprojectionoftheline ontheplane
x+2y+z=12.
Ans.:
24. Provethatthepointwhichtheline meetstheplane2x+35y−39z+12=0is
equidistantfromtheplanes12x−15y+16z=28and6x+6y−7z=8.
25. Findtheequationoftheprojectionofthestraightline ontheplanex+y+2z=5
insymmetricalform.
Ans.:
26. Provethattwolinesinwhichtheplanes3x−7y−5z=1and5x−13y+3z+2=0cuttheplane8x−11y+2z=0includearightangle.
27. Reducetosymmetricalformthelinegivenbytheequationsx+y+z+1=0,4x+y−2z+2=0.Hencefindtheequationoftheplanethrough(1,1,1)andperpendiculartothegivenline.
Ans.:
28. Showthatthelinex+2y−z−3=0,x+3y−z−4=0isparalleltothexz–planeandfindthecoordinatesofthepointwhereitmeetsyz–plane.
Ans.:(0,1,−1)
29. Findtheanglebetweenthelinesx−2y+z=0,x+y−z−3=0,andx+2y+z−5=0,8x+12y+5z=0.
Ans.:
30. Findtheequationoftheplanepassingthroughtheline andparalleltotheline
.
Ans.:11x+2y−5z+6=0
31. Theplane meetstheaxesinA,BandC.Findthecoordinatesoftheorthocentreofthe
ΔABC.
Ans.:
32. TheequationtoalineABare .ThroughapointP(1,2,3),PNisdrawnperpendicularto
ABandPQisdrawnparalleltotheplane2x+3y+4z=0tomeetABinQ.FindtheequationsofPNandPQandthecoordinatesofNandQ.
Ans.:
ILLUSTRATIVEEXAMPLES(COPLANARLINESANDSHORTESTDISTANCE)
Example13.18
Provethatthelines and arecoplanarandfind
theequationoftheplanecontinuingthesetwolines.
Solution
(−1,−10,1)isapointonthefirstlineand−3,8,2arethedirectionratiosofthefirstline.(−3,−1,4)isapointonthesecondlineand−4,7,1arethedirection
ratiosofthesecondline.Ifthelinesarecoplanarthen
Therefore,thetwolinesarecoplanar.Theequationoftheplanecontainingthe
linesis
Example13.19
Showthatthelines and intersect.Findthepointof
intersectionandtheequationoftheplanecontainingthesetwolines.
Solution
Thetwogivenlinesare
Anypointonthefirstlineis(−3r−1,2r+3,r−2).Anypointonthesecondlineis(r1,−3r1+7,2r1−7).Ifthetwolinesintersectthenthetwopointsareoneandthesame.
Solving(13.60)and(13.61),wegetr=−1andr1=2.Thesevaluessatisfyequation(13.59).Thepointofintersectionis(2,1,−3).Theequationoftheplanecontainingthetwolinesis
Example13.20
Showthatthelines andx+2y+3z−8=0,2x+3y+4z−11=0
arecoplanar.Findtheequationoftheplanecontainingthesetwolines.
Solution
Thetwolinesare
Anyplanecontainingthesecondlineis
Ifthelinegivenby(13.62)liesonthisplanethenthepoint(−1,−1,−1)alsoliesontheplane.
Theequationoftheplane(13.64)is
Alsothenormaltothisplaneshouldbeperpendiculartotheline(13.62).Thedirectionratiosofthenormaltotheplaneare4,1,−2.Thedirectionratiosoftheline(13.62)are1,2,3.Alsoll1+mm1+nn1=4+2−6=0whichistrue.Hence,theplanecontainingthetwogivenlinesis4x+y−2z+3=0.Anypointonthefirstlineis(r−1,2r−1,3r−1).Ifthetwogivenlinesintersectatthispointthenitshouldlieonthesecondlineandhenceontheplanex+2y+3z−8=0.
Therefore,thepointofintersectionofthetwogivenlinesis(0,1,2).
Example13.21
Showthatthelinesx+2y+3z−4=0,2x+3y+4z−5=0and2x+3y+3z−5=0,3x−2y+4z−6=0arecoplanarandfindtheequationoftheplanecontainingthetwolines.
Solution
Letusexpressthefirstlineinsymmetricalform.Letl,m,nbethedirectioncosinesofthefirstline.Thenthislineisperpendiculartothenormalsoftheplanesx+2y+3z−4=0and2x+3y+4z−5=0.
Solving,weget
Therefore,thedirectionratiosofthefirstlineare1,−2,1.TofindapointonthefirstlineletusfindwherethislinemeetstheXOYplane
(i.e.)z=0.
Solvingthesetwoequationswegetthepointas(−2,3,0).Therefore,theequationsofthefirstlineare
Anyplanecontainingthesecondlineis
Iftheplanecontainsthesecondlinethenthepoint(−2,3,0)shouldlieontheplane(13.67).
Hencetheequationsoftheplane(13.67)becomes
Alsoitshouldsatisfythecondition.Thatthenormaltotheplaneshouldbeperpendiculartotheline(13.66).Thedirectionratiosofthenormaltotheplane(13.68)are1,1,1.Thedirectionratiosofthelineare1,−2,1.Also1−2+1=0whichis
satisfied.Hencetheequationoftherequiredplaneisx+y+z−1=0.
Example13.22
Provethatthelinesx=ay+b=cz+dandx=αy+β=γz+δarecoplanarif(r−c)(αβ−bd)−(α−a)(αδ−δγ)=0.
Solution
Firstletusexpressthegivenlinesinsymmetricalform.Thetwogivenlines
Thentwolinesarecoplanarif
Example13.23
Provethatthelinesa1x+b1y+c1z+d1=0=a2x+b2y+c2z+d2anda3x+b3y
+c3z+d3=0=a4x+b4y+c4z+d4arecoplanarif .
Solution
Letthetwolinesintersectat(x1,y1,z1).Then(x1,y1,z1)shouldlieontheplanescontainingtheselines.
Eliminating(x1,y1,z1)fromtheaboveequationsweget
Thisistherequiredcondition.
Example13.24
Findtheshortestdistanceandtheequationtothelineofshortestdistance
betweenthetwolines and .
Solution
Thetwogivenlinesare and .
ThecoordinatesofanypointPonthefirstlineare(3r−7,4r−4,−2r−3).ThecoordinatesofanypointQonthesecondlineare(6r1+21,−4r1−5,−r+
2).ThedirectionratiosofthelinePQare3r−6r1−28,4r+4r1+1,−2r+r1−
5.IfPQisthelineoftheshortestdistancethenthetwolinesareperpendicular.
Thedirectionratiosofthetwolinesare3,4,−2and6,−4,−1.Then3(3r−6r1−28)+4(4r+4r1+1)−2(−2r+r1−5)=0and6(3r−6r1−2r)−4(4r+4r1+1)−1(−2r+r1−8)=0
Solvingforrandr1,weget
ThecoordinatesofPandQaregivenbyP(−1,4,−7)andQ(3,7,5).
Theequationsofthelineoftheshortestdistanceare (i.e.)
Example13.25
Showthattheshortestdistancebetweenz–axisandthelineofintersectionofthe
plane2x+3y+z−1=0with3x+2y+z−2=0is units.
Solution
Theequationsoftheplanecontainingthegivenlineis
Thedirectionratiosofthenormaltothisplaneare2+3λ,3+2λ,4+λ.Thedirectionratiosofthez-axisare0,0,1.Ifz-axisisparalleltothelinethen
0(2+3λ),0(3+2λ)+1(4+λ)=0.
∴λ=−4Therefore,theequationoftheplane(13.69)is2x+3y+4z−1−4(3x+2y+z−2)=0
Example13.26
Findthepointsonthelines and whicharenearest
toeachother.Hencefindtheshortestdistancebetweenthelinesandalsoitsequation.
Solution
Thegivenlinesare
Anypointontheline(13.71)isP(3r+6,−r+7,r+4).Anypointontheline(13.72)isQ(−3r1,2r1−9,4r1+2).ThedirectionratiosofPQare(3r+3r1+6,−r−2r1+16,r−4r1+2).SincePQisperpendiculartothetwogivenlines.
Therefore,thepointsPandQare(3,8,3)and(−3,−7,6).TheSDisthedistancePQ.
ThedirectionratiosofPQare6,15,−3(i.e.)2,5,−1.Pis(3,8,3).
Therefore,theequationsofthelineofSDare .
Example13.27
Findtheshortestdistancebetweenthelines and
.Findalsotheequationofthelineoftheshortestdistance.
Solution
Letl,m,nbethedirectionratiosofthelineoftheSD.Sinceitisperpendiculartoboththelines
Solvingforl,m,n,weget
ThedirectionratiosofthelineofSDare2,3,6.Thedirectioncosinesofthe
lineofSDare
ThelengthofthelineoftheSD=|(x1−x2)l+(y1−y2)m+(z1−z2)n|where(x1,y1,z1)and(x2,y2,z2)arethedirectioncosinesofthelineofSD.
TheequationoftheplanecontainingthefirstlineandthelineofSDis
TheequationoftheplanecontainingthesecondlineandthelineofSDis
Therefore,theequationsofthelineofSDare117x+4y+71z−490=0,63x−28y+7z−238=0.
Example13.28
If2distheshortestdistancebetweenthelinesx=0, andy=0,
thenprovethat .
Solution
Thetwogivenlinesare
Theequationofanyplanecontainingthefirstlineis
Theequationofthesecondlineinsymmetricalformis
Theplanegivenbyequation(13.75)isparalleltotheline(13.76).If
Hencefrom(13.75),theequationoftheplanecontainingline(13.73)and
paralleltotheplane(13.74)is .
ThentheSDbetweenthegivenlines=theperpendiculardistancefromanypointontheline(13.74)totheplane(13.75).(0,0,−c)isapointontheline(13.76).
Example13.29
Showthattheshortestdistancebetweenanytwooppositeedgesofthetetrahedronformedbytheplanesy+z=0,z+x=0,x+y=0andx+y+z=a
is andthethreelinesoftheshortestdistanceintersectatthepointx+y+z=
a.
Solution
Theequationsoftheedgedeterminedbytheplanesy+z=0,z+x=0is
Theequationoftheoppositeedgesarex+y=0,x+y+z=a
Letl,m,nbethedirectioncosinesofthelineoftheSDbetweentwolines.Thenl+m−n=0,l−m+0.n=0.
Solvingforl,m,nweget .
Therefore,thedirectioncosinesofthelineoftheSDare .
Theequationoftheplanecontainingtheedgegivenby(13.77)andthelineof
theSDis .
Therefore,theequationsoftheSDaregivenby
Thislinepassesthroughthepoint(a,a,a).Similarly,bysymmetrywenotethattheothertwolinesofSDalsopassthroughthepoint(a,a,a).
Example13.30
AsquareABCDofdiagonal2aisfoldedalongthediagonalAC,sothattheplanesDAC,BACareatrightangles.FindtheshortestdistancebetweenDCandAB.
Solution
Letabethesideofthesquare.LetustakeOB,OC,ODastheaxesofcoordinates.ThecoordinatesofB,C,DandAare(a,0,0),(0,a,0),(0,0,a),(0,0,−a).
TheequationsofABare
TheequationsofCDare
Theequationsoftheplanepassingthroughthestraightline(13.80)andparallel
to(13.81)is .
Therefore,therequiredshortestdistance=perpendicularfromthepoint(0,a,0)totheplane(13.82).
Example13.31
Provethattheshortestdistancebetweenthediagonalofrectangular
parallelepipedandtheedgenotmeetingitis wherea,b,c
aretheedgesoftheparallelepiped.
Solution
LetOA,OBandOCbethecoterminousedgesofarectangularparallelepiped.ThediagonalsareOO′,AA′,BB′andCC′.ThecoordinatesofO′are(a,b,c).ThecoordinatesofBandC′are(a,0,0)and(a,b,0).
TheequationsofOO′are
TheequationsofBCare .
Letl,m,nbethedirectioncosinesofthelineoftheSD.Then
Hence,l,m,nare−c,o,a.
ThedirectioncosinesofthelineoftheSDare
SimilarlywecanprovethattheothertwoSDare, and .
Exercises3
1. Provethatthelines and arecoplanarandfindtheequationof
theplanecontainingtheline.Ans.:x−y+z=0
2. Provethatthelines and intersect.Findthepointof
intersectionandtheplanecontainingtheline.
Ans.:
3. Showthatthelines and intersectandfindtheequationof
theplanecontainingthelines.Ans.:(5,−7,6)
4. Provethattheline and arecoplanar.Findalsothepointof
intersectionandtheequationoftheplanethroughthem.Ans.:(−1,5,8),4x−11y+7z+3=0
5. Showthatthelines and arecoplanar.Findtheequationof
theplanecontainingtheline.Ans.:x−2y+z=0
6. Showthatthelines and arecoplanar.Findthepointof
intersectionandtheequationoftheplanecontainingthem.Ans.:(1,3,2),17x−47y−24z+172=0
7. Showthatthelines and arecoplanarandfindtheequation
oftheplanecontainingthem.Ans.:x−2y+z=0
8. Showthatthelines and arecoplanarandfindthe
equationoftheplanecontainingthem.Ans.:6x−5y−z=0
9. Showthatthelines and intersectandfindtheequationof
theplanecontainingtheselines.
Ans.:
10. Showthatthelines and3x+2y+z−2=0,x−3y+2z−13=0intersect.
Findalsotheequationoftheplanecontainingthem.
Ans.:(−1,2,3),6x−5y−z=0.
11. Showthatthelinesx−3y+2z+4=0,2x+y+4z+1=0and3x+2y+5z−1=0,2y+z=0arecoplanar.Findtheirpointofintersectionandtheequationoftheplanecontainingtheselines.
Ans.:(3,1,−2),3x+4y+6z−1=0.
12. Showthatthelinesx+y+z−3=0,2x+3y+4z−5=0and4x−y+5z−7=0,2x−5y−z−3=0arecoplanar.Findtheequationoftheplanecontainingtheselines.
Ans.:x+2y+3z−2=0.
13. Showthatthelines7x−4y+7z+16=0,4x+3y−2z+3=0andx−3y+4z+6=0,x−y+z+1=0arecoplanar.
14. Showthatthelines7x−2y−2z+3=0,9x−6y+3=0and5x−4y+z=0,6y−5z=0arecoplanar.Findtheequationoftheplaneinwhichtheylie.
Ans.:x−2y+z=0
15. Showthatthelines andx+2y+z+2=0,4x+5y+3z+6=0arecoplanar.
Findthepointofintersectionofthesetwolines.
Ans.:
16. Showthatthelines andx+2y+3z−14=0,3x+4y+5z−26=0are
coplanar.Findtheirpointofintersectionandtheequationoftheplanecontainingthem.Ans.:(1,2,3),11x+2y−7z+6=0.
17. Showthatthelines3x−y−z+2=0,x−2y+3z−6=0and3x−4y+3z−4=0,2x−2y+z−1=0arecoplanar.Findtheirpointofintersectionandtheequationoftheplanecontainingtheselines.
Ans.:(1,2,3),x−z+2=0
18. Showthatthelines2x−y−z−3=0,x−3y+2z−4=0andx−y+z−2=0,4x+y−6z−3=0arecoplanarandfindtheequationoftheplanecontainingthesetwolines.
Ans.:(1,−1,0),x−z−1=0
19. Showthatthelinesx+2y+3z−4=0;2x+3y+4z−5=0and2x−3y+3z−5=0,3x−2y+4z−6=0arecoplanar.Findtheequationoftheplanecontainingthesetwolines.
Ans.:x+y+z−1=0
20. Showthattheequationoftheplanethroughtheline andwhichisperpendiculartothe
planecontainingthelines and is(m−n)x+(n−l)y+(l−m)z=0.
21. Provethatthelines andax+by+cz+d=0,a1x+b1y+c1z+d1=0,are
coplanarif .
22. Showthatthelines and arecoplanarif
.
23. A,A′;B,B′andC,C′arepointsontheaxes,showthatthelinesofintersectionoftheplanes(A′BC,AB′C′),(B′CA,BC′A′)and(C′AB,CA′B′)arecoplanar.
24. Findtheshortestdistancebetweenthelines and andalsothe
equationsofthelineoftheSD.
Ans.: ,4x+y−5z=0,9x+y−8z−31=0
25. Findtheshortestdistancebetweenthelines and andfind
theequationofthelineoftheshortestdistance.
Ans.: ,4x−5y−17z+79=0,22x−5y+19z−83=0
26. Findtheshortestdistancebetweenthelines and Findalso
theequationofthelineofSDandthepointswherethelineofSDintersectthetwogivenlines.
Ans.: ,(3,5,7),(–1,–1,–1)
27. Showthattheshortestdistancebetweenz–axesandthelineofintersectionoftheplane2x+3y+
4z−1=0with3x+2y+z−2=0is .
28. Showthattheshortestdistancebetweenthelines and is
anditsequationare11x+2y−7z+6=0,7x+y−3z+7=0.
29. Findthelengthoftheshortestdistancebetweenthelines and2x+3y−6z−6=0,
3x−2y−z+5=0.
Ans.:
30. Findtheshortestdistancebetweenz–axisandthelineax+by+cz+d=0,a′x+b′y+c′z+d′=0.
Ans.:
31. Findtheshortestdistancebetweenanedgeofacubeandadiagonalwhichdoesnotmeetit.
Ans.:
32. Alinewithdirectioncosinesproportionalto1,7,−5isdrawntointersectthelines
and .Findthecoordinatesofthepointofintersectionand
thelengthinterceptedonit.
Ans.:(2,8,−3),(0,1,2),
33. Alinewithdirectioncosinesproportionalto2,7,−5isdrawntointersectthelines
Findthecoordinatesofthepointsofintersectionandthe
lengthinterceptedonit.
Ans.:(2,8,−3),(0,1,2);
34. Thetwolines and arecutbyathirdlinewhose
directioncosinesareλ,μ,γ.Showthatthelengthinterceptedonthethirdlineisgivenby
÷ andshowthatthelengthoftheshortestdistanceis
35. Thelengthsoftwooppositeedgesofatetrahedronarea,b,c;theshortestdistanceisequaltod
andtheanglebetweenthemisθ.Provethatthevolumeofthetetrahedronis abdsinθ.
36. Showthattheequationoftheplanecontainingthelinex=0, andparalleltotheliney=
0, is .Ifdistheshortestdistancebetweenthelinesthenshowthat
.
37. Showthattheshortestdistancebetweenthelinesy=az+b,z=αx+βandy=a′z+b,z=α′x+
β′yis .
38. Findtheshortestdistancebetweenthelinesx=2z+3,y=3z+4andx=4z+5,y=5z+6.Whatconclusiondoyoudrawfromyouranswer?
Ans.:Zero;Coplanarlines
Chapter14
Sphere
14.1DEFINITIONOFSPHERE
Thelocusofamovingpointinspacesuchthatitsdistancefromafixedpointisconstantiscalledasphere.Thefixedpointiscalledthecentreofthesphere.Theconstantdistanceiscalledtheradius.
14.2THEEQUATIONOFASPHEREWITHCENTREAT(a,b,c)ANDRADIUSr
LetP(x,y,z)beanypointonthesphere.LetC(a,b,c)bethecentre.
Then,
Thisistheequationoftherequiredsphere.
Showthattheequationx2+y2+z2+2ux+2vy+2wz+d=0alwaysrepresentsasphere.Finditscentreandradius.
Addu2+v2+w2tobothsides.
Thisequationshowsthatthisisthelocusofapoint(x,y,z)movingfromthe
fixedpoint(–u,–v,–w)keepingaconstantdistance fromit.
Therefore,thelocusisaspherewhosecentreis(–u,–v,–w)andwhoseradius
is .
Note14.2.1:Ageneralequationofseconddegreeinx,y,zwillrepresentasphereif(i)coefficientsofx2,y2,z2arethesameand(ii)thecoefficientsofxy,yz,zxarezero.
14.3EQUATIONOFTHESPHEREONTHELINEJOININGTHEPOINTS(x1,y1,z1)AND(x2,y2,z2)ASDIAMETER
Findtheequationofthesphereonthelinejoiningthepoints(x1,y1,z1)and(x2,y2,z2)astheextremitiesofadiameter.
A(x1,y1,z1)andB(x2,y2,z2)betheendsofadiameter.Let(x,y,z)beanypointonthesurfaceofthesphere.Then∠APB=90°Therefore,APisperpendiculartoBP.ThedirectionratiosofAParex–x1,y–y1,z–z1.ThedirectionratiosofBParex–x2,y–y2,z–z2.SinceAPisperpendiculartoBP,
Thisistheequationoftherequiredsphere.
14.4LENGTHOFTHETANGENTFROMP(x1,y1,z1)TOTHESPHEREx2+y2+z2+2ux+2vy+
2wz+d=0
FindthelengthofthetangentfromP(x1,y1,z1)tothespherex2+y2+z2+2ux+2vy+2wz+d=0.
Thecentreofthesphereis(–u,–v,–w).
Theradiusofthesphereis .
Note14.4.1:IfPT2>0,thepointPliesoutsidethesphere.IfPT2=0,thenthepointPliesonthesphere.IfPT2<0,thenthepointPliesinsidethesphere.
14.5EQUATIONOFTHETANGENTPLANEAT(x1,y1,z1)TOTHESPHEREx2+y2+z2+2ux+
2vy+2wz+d=0
Findtheequationofthetangentplaneat(x1,y1,z1)tothespherex2+y2+z2
+2ux+2vy+2wz+d=0.
Thecentreofthesphereis(–u,–v,–w).P(x1,y1,z1)isapointonthesphereandtherequiredplaneisatangentplane
tothesphereatP.Therefore,thedirectionratiosofCParex1+u,y1+v,z1+w.
Therefore,theequationofthetangentplaneat(x1,y1,z1)is(x1+u)(x–x1)+(y1+v)(y–y1)+(z1+w)(z–z1)=0.
Addingux1+vy1+wz1+dtobothsides,weget
Therefore,theequationofthetangentplaneat(x1,y1,z1)isxx1+yy1+zz1+u(x+x1)+v(y+y1)+w(z+z1)+d=0.
14.6SECTIONOFASPHEREBYAPLANE
LetCbethecentreofthesphereandPbeanypointonthesectionofthespherebytheplane.DrawCNperpendiculartotheplane.ThenNisthefootoftheperpendicular
fromPontheplanesection.JoinCP.SinceCNisperpendiculartoNP,CNPisarightangledtriangle.
SinceCPandCNareconstants,NP=constantshowsthatthelocusofPisacirclewithcentreatNandradiusequaltoNP.
Note14.6.1:Iftheradiusofthecircleislessthantheradiusofthespherethenthecircleiscalledasmallcircle.Inotherwords,acircleofthespherenotpassingthroughthecentreofthesphereiscalledasmallcircle.
Note14.6.2:Iftheradiusofthecircleisequaltotheradiusofthespherethenthecircleiscalledagreatcircleofthesphere.Inotherwords,acircleofthespherepassingthroughthecentreofthesphereiscalledagreatcircle.
14.7EQUATIONOFACIRCLE
Thesectionofaspherebyaplaneisacircle.Supposetheequationofthesphereis
andtheplanesectionis
Thenanypointonthecirclelieonthesphere(14.1)aswellastheplanesection(14.2).Hence,theequationsofthecircleofthespherearegivenbyx2+y2+z2+2ux+2vy+2wz+d=0andax+by+cz+k=0.
14.8INTERSECTIONOFTWOSPHERES
Thecurveofintersectionoftwospheresisacircle.
Thecurveofintersectionoftwospheresisacircle.
Letthetwospheresbe
Equation(14.3)isalinearequationinx,y,zandthereforerepresentsaplaneandthisplanepassesthroughthepointofintersectionofthegiventwospheres.Inaddition,weknowthatsectionofthespherebyaplaneisacircle.Hence
thecurveofintersectionofthespheresisgivenbyS1−S2=0.
14.9EQUATIONOFASPHEREPASSINGTHROUGHAGIVENCIRCLE
LetthegivencirclebeS=x2+y2+z2+2ux+2vy+2wz+d=0and
ConsidertheequationS+λP=0.
Thisequationrepresentsasphere.Suppose(x1,y1,z1)isapointonthegivencircle.Then
Equations(14.6)and(14.7)showthatthepoint(x1,y1,z1)liesonthespheregivenbyequation(14.5).Since(x1,y1,z1)isanarbitrarypointonthecircle,itfollowsthateverypoint
onthecircleisapointonthespheregivenby(14.5).Henceequation(14.5)representstheequationofaspherepassingthroughthe
circle(14.4).
14.10CONDITIONFORORTHOGONALITYOFTWOSPHERES
Letthetwogivenspheresbe
ThecentresofthespheresareA(–u,–v,–w)andB(–u1,–v1,–w1).Theradius
Twospheresaresaidtobeorthogonal,ifthetangentplanesatthispointofintersectionareatrightangles.
(i.e.)Theradiidrawnthroughthepointofintersectionareatrightangles.
Thisistherequiredcondition.
14.11RADICALPLANE
Thelocusofapointwhosepowerswithrespecttotwospheresareequaliscalledtheradicalplaneofthetwospheres.
14.11.1ObtaintheEquationstotheRadicalPlaneofTwoGivenSpheres
Letthetwogivenspheresbe
Let(x1,y1,z1)beapointsuchthatthepowerofthispointwithrespecttospheres(14.10)and(14.11)beequal.Then
Thelocusof(x1,y1,z1)is
Thisisalinearequationinx,yandzandhencethisequationrepresentsaplane.Henceequation(14.12)istheequationtotheradicalplaneofthetwogiven
spheres.
Note14.11.1.1:Whentwospheresintersect,theplaneoftheirintersectionistheradicalplane.
Note14.11.1.2:Whenthetwospherestouch,thecommontangentplanethroughthepointofcontactistheradicalplane.
14.11.2PropertiesofRadicalPlane
1. Theradicalplaneoftwospheresisperpendiculartothelinejoiningtheircentres.Proof:Lettheequationsofthetwospheresbe
Thecentresofthetwospheresare
C1(–u1,–v1,–w1)andC2(–u2,–v2,–w2).Thedirectionratiosofthelineofcentresareu1–u2,v1–v2,w1–w2.
Theradicalplaneofspheres(14.13)and(14.14)is2(u1–u2)x+2(v1–v2)y+2(w1–w2)z+(d1–d2)=0.Thedirectionratiosofthenormaltotheplaneareu1–u2,v1–v2,w1–w2.Therefore,thelineofcentreisparalleltothenormaltotheradicalplane.Hence,theradicalplaneoftwospheresisperpendiculartothelinejoiningthecentres.
2. Theradicalplanesofthreespherestakeninpairspassthroughaline.Proof:LetS1=0,S2=0,S3=0betheequationsofthethreegivenspheresineachofwhichthe
coefficientsofx2,y2andz2areunity.ThentheequationsoftheradicalplanestakeninpairsareS1–S2=0,S2–S3=0,S3–S1=0.TheseequationshowthattheradicalplanesofthethreespherespassthroughthelineS1=S2=
S3.Hencetheresultisproved.
Note14.11.2.1:Thelineofconcurrenceofthethreeradicalplanesiscalledradicallineofthethreespheres.
3. Theradicalplanesoffourspherestakeninpairsmeetinapoint.Proof:LetS1=0,S2=0,S3=0andS4=0betheequationsofthefourgivenspheres,ineachofwhich
thecoefficientsofx2,y2,z2areunity.
Thentheequationsoftheradicalplanestakentwobytwoare
TheseequationsshowthattheradicalplanesofthefourspheresmeetinatapointgivenbyS1=S2=S3=S4.
Note14.11.2.2:Thepointofconcurrenceoftheradicalplanesoffourspheresiscalledtheradicalcentreofthefourspheres.
14.12COAXALSYSTEM
Definition14.12.1:Asystemofspheresissaidtobecoaxalifeverypairofspheresofthesystemhasthesameradicalplane.
14.12.1GeneralEquationtoaSystemofCoaxalSpheres
LetS=x2+y2+z2+2ux+2vy+2wz+d=0andS1=x2+y2+z2+2u′x+2v′y+2w′z+d′=0betheequationofanytwospheres.Nowconsidertheequation
Nowconsidertheequation
whereλisaconstant.Clearlythisequationrepresentsasphere.Considertwodifferentspheresofthissystemfortwodifferentvaluesofλ.
Thecoefficientsofx2,y2,z2termsin(14.15)are1+λ.
representtwospheresofthesystemwithunitcoefficientsforx2,y2,z2terms.Therefore,theequationoftheradicalplaneof(14.18)and(14.19)is
Sinceλ2≠λ1,S–S′=0whichistheequationtotheradicalplaneofspheres(14.16)and(14.17).Sincethisequationisindependentofλ,everypairofthesystemofspheres
(14.15)hasthesameradicalplane.Henceequation(14.15)representsthegeneralequationtothecoaxalsystemofthespheres.
14.12.2EquationtoCoaxalSystemistheSimplestForm
Inacoaxalsystemofspheres,thelineofcentresisnormaltothecommonradicalplane.
Therefore,letuschoosethex-axisasthelineofcentresandthecommonradicalplaneastheyz-plane,thatis,(x=0).Lettheequationtoasphereofthecoaxalsystembex2+y2+z2+2ux+2vy+
2wz+d=0.Sincethelineofcentresisthex-axis,isyandzcoordinatesarezerov=0,w=
0.Thentheequationoftheabovespherereducestotheformx2+y2+z2+2ux+
d=0.Letusnowconsidertwosphereofthissystemsay,x2+y2+z2+2ux+d=0
andx2+y2+z2+2u1x+d1=0.Theradicalplaneofthesetwospheresis
Buttheequationoftheradicalplaneisx=0.Therefore,from(14.20),d–d1=0ord1=d
Hencetheequationtoanysphereofthecoaxalsystemisoftheformx2+y2+z2
+2λx+d=0whereλisavariableanddisaconstant.
14.12.3LimitingPoints
Limitingpointsaredefinedtobethecentresofpointspheresofthecoaxalsystem.Lettheequationtoacoaxalsystembe
Centreis(–λ,0,0)andradiusis
Forpointsphereradiusiszero.
Therefore,thelimitingpointsofthesystemofspheresgivenby(14.21)are(
,0,0)and(– ,0,0).
Note14.12.3.1:Limitingpointsarerealorimaginaryaccordingasdispositiveornegative.
14.12.4IntersectionofSpheresofaCoaxalSystem
Lettheequationtoacoaxalsystemofspherebex2+y2+z2+2λx+d=0.Nowconsidertwospheresofthesystemsay
Nowconsidertwospheresofthesystemsay
TheintersectionofthesetwospheresisS1–S2=0.
(i.e.)2(λ1–λ2)x=0(i.e.)x=0sinceλ1≠λ2substitutingx=0in(14.22)or(14.23)weget,
Therefore,thisequationisacircleintheyz-planeandalsoitisindependentofλ.Henceeverysphereofthesystemmeetstheradicalplanewithsamecircle.
Note14.12.4.1:Thiscircleiscalledthecommoncircleofthecoaxalsystem.
Note14.12.4.2:Ifd<0,thecommoncircleisrealandthesystemofspheresaresaidtobeintersectingtype.Ifd=0,thecommoncircleisapointcircleandinthiscaseanytwospheresofthesystemtoucheachother.
Ifd>0,thecommoncircleisimaginaryandthespheresaresaidtobeofnon-intersectingtype.
ILLUSTRATIVEEXAMPLES
Example14.1
Findtheequationofthespherewithcentreat(2,–3,–4)andradius5units.
Solution
Theequationofthespherewhosecentreis(a,b,c)andradiusris(x–a)2+(y–b)2+(z–c)2=r2.Therefore,theequationofthespherewhosecentreis(2,–3,–4)andradius5is(x–2)2+(y+3)2+(z–4)2=52.
Example14.2
Findthecoordinateofthecentreandradiusofthesphere16x2+16y2+16z2–16x–8y–16z–35=0.
Solution
Theequationofthesphereis16x2+16y2+16z2–16x–8y–16z–35=0.
Dividingby16,
Centreofthesphereis .
Example14.3
Findtheequationofthespherewiththecentreat(1,1,2)andtouchingtheplane2x–2y+z=5.
Solution
Theradiusofthesphereisequaltotheperpendiculardistancefromthecentre(1,1,2)ontheplane2x–2y+z–5=0.
Theequationofthespherewithcentreat(1,1,2)andradius1unitis(x–1)2+(y–1)2+(z–2)2=1.
(i.e.)x2+y2+z2–2x–2y–4z+5=0
Example14.4
Findtheequationofthespherepassingthroughthepoints(1,0,0),(0,1,0),(0,0,1)and(0,0,0).
Solution
Lettheequationofthespherebex2+y2+z2+2ux+2vy+2wz+d=0.Thispassesthroughthepoints(1,0,0),(0,1,0),(0,0,1)and(0,0,0).
Theequationofthesphereisx2+y2+z2–x–y–z=0.
Example14.5
Findtheequationofthespherewhichpassesthroughthepoints(1,0,0),(0,1,0)and(0,0,1)andhasitscentreontheplanex+y+z=6.
Solution
Lettheequationofthespherebex2+y2+z2+2ux+2vy+2wz+d=0.Thisspherepassesthroughthepoints(1,0,0),(0,1,0)and(0,0,1).
Thecentreofthesphereis(–u,–v,–w).Thisliesontheplanex+y+z–6=0.
Theequationofthesphereisx2+y2+z2–4x–4y–4z+4=0.
Example14.6
Findtheequationofthespheretouchingtheplane2x+2y–z=1andconcentricwiththesphere2x2+2y2+2z2+x+2y–z=0.
Solution
Centreis .
Thespheretouchestheplane2x+2y–z–1=0.
Theequationofthesphereis
Example14.7
Findtheequationofthespherewhichpassesthroughthepoints(2,7,–4)and(4,5,–1)hasitscentreonthelinejoiningthethesetwopointsasdiameter.
Solution
Aliter:Thetwogivenpointsaretheextremitiesofadiameterofthesphere.
Therefore,theequationofthesphereis
Example14.8
Theplane cutsthecoordinateaxesinA,BandC.Findtheequationof
thespherepassingthroughA,B,CandO.Findalsoitscentreandradius.
Solution
Theplane cutsthecoordinatesofA,BandC.ThecoordinatesofA,B
andCare(a,0,0),(0,b,0)and(0,0,c).
LettheequationofthespherepassingthroughA,BandCbex2+y2+z2+2ux+2vy+2wz+d=0.SincethispassesthroughoriginO,d=0.SincethispassesthroughA,Band
C.
Hencetheequationofthesphereisx2+y2+z2–2ax–2by–2cz=0.
Centreofthesphereis(a,b,c)andradiusofthesphere=
Example14.9
Findtheequationofthespherecircumscribingthetetrahedronwhosefacesare
and
Solution
Thefacesofthetetrahedronare
Noweasilyseenthattheverticesofthetetrahedronare(0,0,0),(a,b,–c),(a,–b,c)and(–a,b,c).Lettheequationofthespherebex2+y2+z2+2ux+2vy+2wz+d=0.Thisspherepassesthroughthepoints(0,0,0),(a,b,–c,),(a,–b,c)and(–a,b,
c).
Adding(14.28)and(14.29),2(a2+b2+c2)+4ua=0
Similarly,
Therefore,theequationofthesphereisx2+y2+z2–(a2+b2+c2)
Example14.10
Asphereisinscribedinatetrahedronwhosefacesarex=0,y=0,z=0and2x+6y+3z=14.Findtheequationofthesphere.Alsofinditscentreandradius.
Solution
Lettheequationofthespherebex2+y2+z2+2ux+2vy+2wz+d=0.Sincethespheretouchestheplanex=0,theperpendiculardistancefromthe
centre(–u,–v,–w)onthisplaneisequaltotheradius.
∴–u=r,–v=r,–w=r.Alsothespheretouchestheplane2x+6y+3z–14=0.
When ,theequationofthesphereis
Forthissphere,centreis andradius= .
When ,
Example14.11
Findtheequationofthespherepassingthroughthepoints(1,0,–1),(2,1,0),(1,1,–1)and(1,1,1).
Solution
Lettheequationofthespherebex2+y2+z2+2ux+2vy+2wz+d=0.Thisspherepassesthrough(1,0,–1),(2,1,0),(1,1,–1)and(1,1,1).
From(14.31),–2+d=–2⇒d=0.Therefore,therequiredequationofthecircleisx2+y2+z2–2x–y=0.
Example14.12
Findtheequationofthespherewhichtouchesthecoordinateaxes,whosecentreliesinthepositiveoctantandhasaradius4.
Solution
Lettheequationofthespherebex2+y2+z2+2xu+2vy+2wz+d=0.
Theequationofthex-axisis .
Anypointonthislineis(t,0,0).Thepointliesonthegivenspheret2+2ut+d=0.
Sincethespheretouchesthex-axisthetworootsofthisequationareequal.
∴4u2–4d=0oru2=d.
Similarly,v2=dandw2=d
Theradiusofthesphereis
Sincethecentreliesonthex-axis,–u=–v=–w=2 .
Therefore,therequiredequationisx2+y2+z2–4 (x+y+z)+8=0.
Example14.13
Findtheradiusandtheequationofthespheretouchingtheplane2x+2y–z=0andconcentricwiththesphere2x2+2y2+2z2+x+2y–z=0.
Solution
Sincetherequiredsphereisconcentricwiththesphere2x2+2y2+2z2+x+2y–
z=0itscentreisthesameasthatofthegivensphere
Centreis .Theradiusoftherequiredsphereisequaltothe
perpendiculardistancefromthispointtotheplane2x+2y–z=0.
Theequationoftherequiredsphereis
Example14.14
Findtheequationofthespherewhichpassesthroughthepoints(1,0,0),(0,2,0),(0,0,3)andhasitsradiusassmallaspossible.
Solution
Lettheequationofthespherebex2+y2+z2+2ux+2vy+2wz+k=0.Thisspherepassesthroughthepoints(1,0,0),(0,2,0)and(0,0,3).
Theradiusofthesphereisgivenbyr2=u2+v2+w2–k.
Therequiredequationofthesphereis
Example14.15
Findtheequationofthespheretangentialtotheplanex–2y–2z=7at(3,–1,–1)andpassingthroughthepoint(1,1,–3).
Solution
TheequationofnormalatAis
Anypointinthislineis(r+3,–2r–1,–2r–1).IfthispointisthecentreofthespherethenCA=CB.
Therefore,centreofthesphereis(0.5,5).
Radius=
Therefore,theequationofthesphereis(x–0)2+(y–5)2+(z–5)2=81.
(i.e.)x2+y2+z2–10y–10z–31=0
Example14.16
Showthattheplane4x–3y+6z–35=0isatangentplanetothespherex2+y2
+z2–y–2z–14=0andfindthepointofcontact.
Solution
Iftheplaneisatangentplanetothespherethentheradiusisequaltotheperpendiculardistancefromthecentreontheplane.
Thecentreofthespherex2+y2+z2–y–2z–14=0is .
Perpendiculardistancefromthecentreontheplaneis
Therefore,theplanetouchesthesphere.Theequationsofthenormaltothe
tangentplaneare
Anypointonthislineis .
Ifthispointliesontheplane4x–3y+6z–35=0then,
Therefore,thepointofcontactis(2,–1,4).
Example14.17
AsphereofconstantradiusrpassesthroughtheoriginOandcutstheaxesinA,BandC.FindthelocusofthefootoftheperpendicularfromOtotheplaneABC.
Solution
Lettheequationofthespherebex2+y2+z2+2ux+2vy+2wz+d=0.Thispassesthroughtheorigin.
∴d=0
ThespherecutstheaxesatA,BandCwhereitmeetsthex-axis.
y=0,z=0∴x2+2ux=0∴x=–2uTherefore,thecoordinatesofAare(–2u,0,0).SimilarlythecoordinatesofBandCareB(0,–2v,0)andC(0,0,–2w).Therefore,theequationsofthesphereisx2+y2+z2–2ux–2vy–2wz=0.Radius=r
TheequationsoftheplaneABCis
Thedirectionratiosofthenormaltothisplaneare .
Theequationsofthenormalare
Let(x,y,z)bethefootoftheperpendicularfromOontheplane.Then(x1,y1,z1)lieson(14.37).
Substitutingin(14.35),
Thepoint(x1,y1,z1)alsoliesontheplane(14.36)
or
Multiplying(14.38)and(14.39),weget
Thelocusof
Example14.18
Asphereofconstantradius2kpassesthroughtheoriginandmeetstheaxesinA,BandC.ShowthatthelocusofthecentroidofthetetrahedronOABCisx2+y2
+z2=k2.
Solution
Lettheequationofthespherebex2+y2+z2+2ux+2vy+2wz+d=0.Thispassesthroughtheorigin.
Sincex≠0,x=–2u.Therefore,thecoordinatesofAare(–2u,0,0).SimilarlythecoordinatesofBandCare(0,–2v,0)and(0,0,–2w).
Let(x1,y1,z1)bethecentroidofthetetrahedronOABC.
Theradiusofthesphereisr.
∴u2+v2+w2=4r2
Using(14.40),
Thelocusof(x1,y1,z1)isthespherex2+y2+z2=r2.
Example14.19
AsphereofconstantradiusrpassesthroughtheoriginandmeetstheaxesinA,BandC.ProvethatthecentroidofthetriangleABCliesonthesphere9(x2+y2
+z2)=4r2.
Solution
Lettheequationofthespherebex2+y2+z2+2ux+2vy+2wz+d=0.Thislinepassingthroughtheorigin.
∴d=0.
Whenthecirclemeetsthex-axis,y=0,z=0
∴x2+2ux=0as
x≠0,x=−2u∴Aisthepoint(–2u,0,0).SimilarlyBandCarethepoints(0,–2v,0)and(0,0,–2w).Alsogiventhe
radiusisr.
Let(x1,y1,z1)bethecentroidofthetriangleABC.
Butthecentroidis
∴from(14.41):
Thelocusof(x1,y1,z1)is9(x2+y2+z2)=4r2.
Example14.20
Aplanepassesthroughthefixedpoint(a,b,c)andmeetstheaxesinA,B,C.
Provethatthelocusofthecentreofthesphereis
Solution
Lettheequationofthespherebex2+y2+z2+2ux+2vy+2wz+d=0.Thispassesthroughtheorigin.
∴d=0Whenthisspheremeetsthex-axis,y=0andz=0.
∴x2+2ux=0.Asx≠0,x=–2u.Therefore,thecoordinatesofAare(–2u,0,0).SimilarlythecoordinatesofBandCare(0,–2v,0)and(0,0,–2w).
TheequationoftheplaneABCis
Thisplanepassesthroughthepoint(a,b,c).
Thelocusofthecentre(–u,–v,–w)is
Example14.21
Findthecentreandradiusofthecirclex2+y2+z2–8x+4y+8z–45=0,x–2y+2z=3.
Solution
Thecentreofthespherex2+y2+z2–8x+4y+8z–45=0is(4,–2,–4).
CNistheperpendicularfromthecentreofthesphereontheplanex–2y+2z=3.
Therefore,theradiusofthecircleis units.TheequationofthelineCNis
Anypointonthislineist+4,–2t–2,2t–4.Thispointisthecentreofthecirclethenthisliesontheplanex–2y+2z–3=0thent+4–2(2t–2)+2(2t–4)–3=0.
Therefore,thecentreofthecircleis
Example14.22
Showthatthecentresofallsectionsofthespherex2+y2+z2=r2byplanesthroughthepoint(α,β,γ)lieonthespherex(x–α)+y(y–β)+z(z–γ)=0.
Solution
Let(x1,y1,z1)beacentreofasectionofthespherex2+y2+z2=r2byaplanethrough(α,β,γ).Thentheequationoftheplaneisx1(x–x1)+y1(y–y1)+z1(z–z1)=0.Thisplanepassesthroughthepoint(α,β,γ).
x1(α–x1)+y1(β–y1)+Z(γ–z1)=0Therefore,thelocusof(x1,y1,z1)isx(α–x)+y(β–y)+z(γ–z)=0.(i.e.)x(x–α)+y(β–y)+z(γ–z)=0whichisasphere.
Example14.23
Findtheequationofthespherehavingthecirclex2+y2+z2=5,x–2y+2z=5foragreatcircle.Finditscentreandradius.
Solution
Anyspherecontainingthegivencircleisx2+y2+z2–5+2λ(x–2y+2z−5)=0.Thecentreofthissphereis(–λ,2λ,–2λ).Sincethegivencircleisagreat
circle,thecentreofthesphereshouldlieontheplanesectionx–2y+2z=5.
Therefore,theequationofthesphereisx2+y2+z2–5– (x–2y+2z–5)=0.
9(x2+y2–z2–5)–10(x–2y+2z–5)=0.
Centreofthesphereis .
Example14.24
Findtheequationsofthesphereswhichpassesthroughthecirclex2+y2+z2=5,x+2y+3z=3andtouchtheplane4x+3y=15.
Solution
Anyspherecontainingthegivencircleisx2+y2+z2–5+λ(x+2y+3z–3)=0.
Centreis .
Ifthespheretouchestheplane4x+3y=15thentheradiusofthesphereisequaltotheperpendiculardistancefromthecentreontheplane.
Therearetwospherestouchingthegivenplanewhoseequationsarex2+y2+z2
–5+2(x+2y+3z–3)=0andx2+y2+z2–5– (x+2y+3z–3)=0
Example14.25
Provethatthecirclesx2+y2+z2–2x+3y+4z–5=0,5y+6z+1=0andx2+y2+z2–3x–4y+5z–6=0,x+2y–7z=0lieonthesamesphereandfinditsequation.
Solution
Theequationofthespherethroughthefirstcircleis
Theequationofthespherethroughthesecondcircleis
Thegivencircleswilllieonthesamesphereifequation(14.42)and(14.43)areidentical.Therefore,comparingequations(14.42)and(14.43)weget,
Thesetwovaluesλandμsatisfy(14.42),theequations(14.45)and(14.46).Hence,thetwogivencircleslieonthesamesphere.Theequationofthe
sphereisx2+y2+z2–3x–4y+5z–6+x+2y–7z=0.
(i.e.)x2+y2+z2–2x–2y–2z–6=0
Example14.26
Theplane meetsthecircleO,A,BandC.Findtheequationsofthe
circumcircleofthetriangleABCandalsofinditscentre.
Solution
TheequationoftheplaneABCis .
Therefore,thecoordinatesofA,BandCare(a,0,0),(0,b,0)and(0,0,c)respectively.AlsoweknowthattheequationofthesphereOABCisx2+y2+z2–ax–by–
cz=0.Therefore,theequationofthecircumcircleofthetriangleABCarex2+y2+z2
–ax–by–cz=0and .
ThecentreofthesphereOABCis
TheequationofthenormalCNis
Anypointonthislineis .
Thus,pointliesontheplane
Hencethecentreofthecircleis
Example14.27
Obtaintheequationstothespherethroughthecommoncircleofthespherex2+y2+z2+2x+2y=0andtheplanex+y+z+4=0whichintersectstheplanex+y=0incircleofradius3units.
Solution
Theequationofthespherecontainingthegivencircleisx2+y2+z2+2x+2y+λ(x+y+z+4)=0.
Centreofthissphereis
CN=PerpendicularfromthecentreContheplanex+y=0.
Therefore,theequationsoftherequiredspheresarex2+y2+z2+2x+2y–2(x+y+z+4)=0andx2+y2+z2+2x+2y+18(x+y+z+4)=0.
Example14.28
Findtheequationofthespherewhichtouchesthespherex2+y2+z2+2x–6y+1=0atthepoint(1,2,–2)andpassesthroughtheorigin.
Solution
Theequationofthetangentplaneat(1,2,–2)isx+2y–2z+(x+1)–3(y+2)+1=0.
Theequationofthespherepassingthroughtheintersectionof(14.48)and(14.49)isx2+y2+z2+2x–6y+1+λ(2x–y–2z–4)=0.Thisspherepassesthroughtheorigin.
Therefore,theequationoftherequiredsphereis4(x2+y2+z2+2x–6y+1)+(2x–y–2z–4)=0.
(i.e.)4(x2+y2+z2)+10x–25y–2z=0
Example14.29
Showthattheconditionforthespherex2+y2+z2+2ux+2vy+2wz+d=0tocutthespherex2+y2+z2+2u1x+2v1y+2w1z+d1=0inagreatcircleis
wherer1istheradiusofthelattersphere.
Solution
TheintersectionofthesetwosphereisS–S1=0.
(i.e.)2(u–u1)x+2(v–v1)y+2(w–w1)z+d–d1=0.
ThecentreofthesphereS1=0is(–u1,–v1,–w1).SinceS1=0cutsS2=0inagreatcircle,thecentreofthesphereliesontheplaneofintersectionS1–S2=0.
Example14.30
Atangentplanetothespherex2+y2+z2=r2makesinterceptsa,bandconthecoordinateaxes.Provethata–2+b–2+c–2=r–2.
Solution
LetP(x1,y1,z1)beapointonthespherex2+y2+z2=r2.
TheequationofthetangentplaneatPisxx1+yy1+zz1=r2.
Therefore,theinterceptsmadebytheplaneonthecoordinateaxesare
Example14.31
Twospheresofradiir1andr2intersectorthogonally.Provethattheradiusofthe
commoncircleis .
Solution
Lettheequationofthecommoncirclebe
Thentheequationofthespherethroughthegivencircleisx2+y2+z2–r2+λz=0whereλisarbitrary.Lettheequationofthetwospheresthroughthegivencirclebex2+y2+z2–r2+λ1z=0andx2+y2+z2–r2+λ2z=0Ifr1andr2aretheradiioftheabovetwospheresthen
Sincethetwospherescutorthogonally.
Eliminatingλ1andλ2from(14.52)and(14.53),weget
Example14.32
Findtheequationofthespherewhichtouchestheplane3x+2y–z+2=0atthepoint(1,–2,1)andcutsorthogonallythespherex2+y2+z2–4x+6y+4=0.
Solution
Lettheequationoftherequiredspherebe
Thisspheretouchestheplane3x+2y–z+2=0at(1,–2,1).Theequationofthetangentplaneat(1,–2,1)isxx1+yy1+zz1+u(x+x1)+v(y+y1)+w(z+z1)+d=0.
Butthetangentplaneisgivenas
Identifyingequations(14.55)and(14.56)weget,
Thesphere(14.54)cutsorthogonallythesphere
Therefore,theequationofthesphereisx2+y2+z2+7x+10y–5z+12=0.
Example14.33
Findtheequationsoftheradicalplanesofthespheresx2+y2+z2+2x+2y+2z+2=0,x2+y2+z2+4y=0andx2+y2+z2+3x–2y+8z+6=0.Alsofindtheradicallineandtheradicalcentre.
Solution
Considertheequations,
Theradicalplaneofthespheres(14.62)and(14.63)isS1–S2=0.
Theradicalplaneofthespheres(14.63)and(14.64)isS2–S3=0.
(i.e.)3x–6y–8z+6=0
Theradicalplaneofthesphere(14.62)andS3isS1–S3=0.
(i.e.)x–4y+6z+4=0Theequationoftheradicallineofthespheresaregivenby
Alsotheradicallineisgivenby
3x–6y+8z+6=0,2x–3y+7z+4=0.Theradicalcentreisthepointofintersectionoftheabovetwolines.Sowehavetosolvetheequations
Solvingtheseequationsweget
Therefore,theradicalcentreis .
Example14.34
Findtheequationofthespherethroughtheoriginandcoaxalwiththespheresx2
+y2+z2=1andx2+y2+z2+x+2y+3z–5=0.
Solution
TheradicalplaneofthetwogivenspheresisS–S1=0.
(i.e.)x+2y+3z–4=0TheequationofanyspherecoaxalwithgivenspheresisS+λP=0.
(i.e.)x2+y2+z2–1+λ(x+2y+3z–4)=0Thisspherepassesthroughtheorigin.
Therefore,theequationoftherequiredsphereisx2+y2+z2–1– (x+2y+3z
–4)=0.
Example14.35
Findthelimitingpointsofthecoaxalsystemofspheresdeterminedbyx2+y2+z2+4x–2y+2z+6=0andx2+y2+z2+2x–4y–2z+6=0.
Solution
Theradicalplaneofthetwogivenspheresis2x+2y+4z=0.Theequationtoanysphereofthecoaxalsystemisx2+y2+z2+4x–2y+2z
+6+λ(x+y+2z)=0.
Thecentreis .
Radiusis
Forlimitingpointofthecoaxalsystemradius=0.
Therefore,thelimitingpointsarethecentresofpointspheresofthecoaxalsystem.Therefore,thelimitingpointsare(–2,1,–1)and(–1,2,1).
Example14.36
Thepoint(–1,2,1)isalimitingpointofacoaxalsystemofspheresofwhichx2
+y2+z2+3x–2y+6=0isamember.Findtheequationoftheradicalplaneofthissystemandthecoordinatesofotherlimitingpoint.
Solution
Thepointgivesbelongingtothecoaxalsystemcorrespondingtothelimitingpoint(–1,2,1)is(x+1)2+(y–2)2+(z–1)2=0.
(i.e.)x2+y2+z2+2x–4y–2z+6=0Twomembersofthesystemofthesystemarex2+y2+3x–3y+6=0andx2+y2+z2+2x–4y–2z+6=0.Theradicalplaneofthecoaxalsystemisx+y+2z=0.Anymemberofthe
systemisx2+y2+2x–4y–2z+6+λ(x+y+2z)=0.
Centreis
Forlimitingpointsradius=0
Whenλ=0centreis(–1,2,1)whichisthegivenlimitingpoint.Whenλ=2,thecentreis(–2,1,–1)whichisotherlimitingpoint.
Example14.37
Showthatthespheresx2+y2+z2=25andx2+y2+z2–24x–40y–18z+225=0touchexternally.Findtheirpointofcontact.
Solution
Thecentreandradiusofsphere(14.65)are
C1(0,0,0),r1=5Centreandradiusofthesphere(14.66)are
Thedistancebetweenthecentres
Hencethetwogivenspherestouchexternally.
Therefore,thepointofcontactdividesthelinesofcentresintheratio4:1Therefore,thecoordinatesofthepointofcontactis
Exercises1
1. Findtheequationofthespherewithi. centreat(1,–2,3)andradius5units.
ii. centreat andradius1unit.
iii. centreat(1,2,3)andradius4units.
2. Findthecoordinatesofthecentreandradiusofthefollowingspheres:
i. x2+y2+z2+2x–4y–6z+15=0
ii. 2x2+2y2+2z2–2x–4y–6z–1=0
iii. ax2+ay2+az2+2ux+2vy+2wz+d=0
3. Findtheequationofthespherewhosecentreisatthepoint(1,2,3)andwhichpassesthroughthepoint(3,1,2).
Ans.:x2+y2+z2–2x–4y–6z+8=0
4. Findtheequationofthespherepassingthroughpoints:i. (0,0,0),(0,1,–1),(–1,2,0)and(1,2,3)ii. (2,0,1),(1,–5,–1),(0,–2,3)and(4,–1,3)iii. (0,–1,2),(0,–2,3),(1,5,–1)and(2,0,1)iv. (–1,1,1),(1,–1,1),(1,1,–1),(0,0,0)
5. Findtheequationofthesphereonthelinejoiningthepoints(2,–3,–1)and(1,–2,–1)attheendsofadiameter.
Ans.:x2+y2+z2–3x+5y+7=0
6. Findtheradiusofthespheretouchingtheplane2x+2y–z–1=0andconcentricwiththesphere
2x2+2y2+2z2+x+2y–z=0.
Ans.: units
7. Findtheequationofthespherepassingthroughthepoints(0,2,3),(1,1,–1),(–5,4,2)andhavingitscentreontheplane3x+4y+2z–6=0.
Ans.:9(x2+y2+z2)+28x+7y–20z–96=0
8. Provethataspherecanbemadetopassthroughthemidpointsoftheedgesofatetrahedronwhose
facesarex=0,y=0,z=0and .Finditsequation.
Ans.:x2+y2+z2–ax–by–cz=0
9. Findtheconditionthattheplanelx+my+nz=pmaytouchthespherex2+y2+z2+2ux+2vy+2wz+d=0.
Ans.:(ul+vm+wn+p)2=(l2+m2+n2)(u2+v2+w2–d)
10. Provethatthespherecircumscribingthetetrahedronwhosefacesarey+z=0,z+x=0,x+y=0
andx+y+z=1isx2+y2+z2–3(x+y+z)=0.11. Apointmovessuchthat,thesumofthesquaresofitsdistancesfromthesixfacesofacubeisa
constant.Provethatitslocusisthespherex2+y2+z2=3(k2–a2).
12. Provethatthespheresx2+y2+z2=100andx2+y2+z2–12x+4y–6z+40=0touchinternallyandfindthepointofcontact.
Ans.:
13. Provethatthespheresx2+y2+z2=25andx2+y2+z2–24x–40y–18z+225=0touchexternally.Findthepointofcontact.
Ans.:
14. Findtheconditionthattheplanelx+my+nz=ptobeatangenttothespherex2+y2+z2=r2.
Ans.:r2(l2+m2+n2)=p2
15. Findtheequationofthespherewhichtouchesthecoordinateplanesandwhosecentreliesinthefirstoctant.
Ans.:x2+y2+z2–2vx–2vy–2vz+2v2=0
16. Findtheequationofthespherewithcentreat(1,–1,2)andtouchingtheplane2x–2y+z=3.
Ans.:x2+y2+z2–2x+2y+z+5=0
17. Findtheequationofthespherewhichhasthepoints(2,7,–4)and(4,5,–1)astheextremitiesofadiameter.
Ans.:x2+y2+z2–6x–12y+5z+47=0
18. Findtheequationofthespherewhichtouchesthethreecoordinateplanesandtheplane2x+y+2z=6andbeinginthefirstoctant.
Ans.:x2+y2+z2–6x–6y–6z+18=0
19. ApointPmovesfromtwopointsA(1,3,4)andB(1,–2,–1)suchthat3.PA=2.PB.Showthatthe
locusofPisthespherex2+y2+z2–2x–4y–16z+42=0.ShowalsothatthisspheredividesAandBinternallyandexternallyintheration2:3.
20. Aplanepassesthroughafixedpoint(a,b,c).Showthatthelocusofthefootoftheperpendicular
toitfromtheoriginisthespherex2+y2+z2–ax–by–cz=0.21. AvariablespherepassesthroughtheoriginOandmeetsthecoordinateaxesinA,BandCsothat
thevolumeofthetetrahedronOABCisaconstant.Findthelocusofthecentreofthesphere.
Ans.:xyz=k2
22. Findtheequationofthesphereonthelinejoiningthepoints:
i. (4,–1,2)and(2,3,6)astheextremitiesofadiameterii. (2,–3,4)and(–5,6,–7)astheextremitiesofadiameter
Ans.:x2+y2+z2–6x–2y–8z+17=0
x2+y2+z2+3x–3y+3z–56=0
23. Aplanepassesthroughafixedpoint(a,b,c)andcutstheaxesinA,BandC.Showthatthelocus
ofcentreofthesphereABCis
24. Asphereofconstantradius2kpassesthroughtheoriginandmeetstheaxesinA,BandC._Prove
thatthelocusofthecentroidofΔABCis9(x2+y2+z2)=a2.
25. Thetangentplaneatanypointofthespherex2+y2+z2=a2meetsthecoordinateaxesatA,BandC.FindthelocusofthepointofintersectionoftheplanesdrawnparalleltothecoordinateplanesthroughA,BandC.
Ans.:x−2+y−2+z–2=a−2
26. OA,OBandOCarethreemutuallyperpendicularlinesthroughtheoriginandtheirdirectioncosinesarel1,m,n;l2,m2,n2andl3,m3,n3.IfOA=a,OB=b,OC=cthenprovethatthe
equationofthesphereOABCisx2+y2+z2–x(al1+bl2+cl3)–y(am1+bm2+cm3)–z(an1+bn2+cn3)=0.
Exercises2
1. Findthecentreandradiusofasectionofthespherex2+y2+z2=1bytheplanelx+my+nz=1.
Ans.:
2. Findtheequationofthespherethroughthecirclex2+y2+z2=5,x+2y+3z=3andthepoint(1,2,3).
Ans.:5(x2+y2+z2)–4x–8y–12z–13=0.
3. Provethattheplanex+2y–z=4cutsthespherex2+y2+z2–x+z–2=0inacircleofradiusunityandfindtheequationofthespherewhichhasthiscircleforoneofitsgreatcircles.
Ans.:x2+y2+z2–2x–2y+2z–2=0
4. Findthecentreandradiusofthecircleinwhichthespherex2+y2+z2=25iscutbytheplane2x+y+2z=9.
Ans.:(2,1,2);4
5. Showthattheintersectionofthespherex2+y2+z2–2x–4y–6z–2=0andtheplanex–2y+
2z–20=0isacircleofradius withitscentreat(2,4,5).
6. Findthecentreandradiusofthecirclex2+y2+z2–2x–4z+1=0,x+2y+2z=11.
Ans.:
7. Provethattheradiusofthecirclex2+y2+z2+x+y+z=4,x+y+z=0is2.
8. Findthecentreandradiusofthecirclex2+y2+z2+2x–2y–4z–19=0,x+2y+2z+7=0.
Ans.:
9. Findtheradiusofthecircle3x2+3y2+3z2+x–5y–2=0,x+y=2.
10. Showthetwocircles2(x2+y2+z2)+8x+13y+17z–17=0,2x+y–3z+1=0andx2+y2+
z2+3x–4y+3z=0,x–y+2z−4=0lieonthesamesphereandfinditsequation.
Ans.:x2+y2+z2+5x–6y+7z–8=0
11. Provethatthetwocirclesx2+y2+z2–2x+3y+4z–5=0,5y+6z+1=0andx2+y2+z2–3x–4y+5z−6=0,x+2y−7=0lieonthesamesphereandfinditsequation.
Ans.:x2+y2+z2–2x–2y–2z–6=0
12. Findtheareaofthesectionofthespherex2+y2+z2+12x–2y–6z+30=0bytheplanex–y+2z+5=0.
Ans.:
13. Findtheequationofthespherewhichhasitscentreontheplane5x+y–4z+3=0andpassing
throughthecirclex2+y2+z2–3x+4y–2z+8=0,4x–5y+3z–3=0.
Ans.:x2+y2+z2+9x–11y+7z–1=0
14. Findtheequationofthespherehavingthecirclex2+y2+z2+10x–4z–8=0,x+y+z–3=0asagreatcircle.
Ans.:x2+y2+z2+6x–4y–3z+4=0
15. Avariableplaneisparalleltoagivenplane andmeetstheaxesatA,BandC.Prove
thatthecircleABCliesonthesurface
16. Findtheequationofthesphereswhichpassthroughthecirclex2+y2+z2–4x–y+6z+12=0,2x+3y–7z=10andtouchtheplanex–2y–2z=1.
Ans.:x2+y2+z2–2x+2y–4z+2=0
x2+y2+z2–6x–4y+10z+22=0
17. Findtheequationofthespherewhichpassthroughthecirclex2+y2+z2=5,x+2y+3z=5andtouchtheplanez=0.
Ans.:x2+y2+z2–2x+y+5z+5=0
5(x2+y2+z2)–(2x–4y+5z+1=0)
18. Findthecentreandradiusofthecirclex2+y2+z2–2x+4y+2z–6=0,x+2y+2z–4=0.
Ans.:(2,0,1),
19. Findtheequationofthespherewhichpassesthroughthepoint(3,1,2)andmeetsXOYplaneinacircleofradius3unitswiththecentreatthepoint(1,–2,0).
Ans.:x2+y2+z2−2x+4y–4z–4=0
20. Findthecentreandradiusofthecirclex2+y2+z2+12x–12y–16z+111=0,2x+2y+z=17.Ans.:(–4,8,9),r=4
21. Findthecentreandradiusofthecirclex2+y2+z2+2x+2y–4z–19=0,x+2y+2z+7=0.
Ans.:
22. Findthecentreandradiusofthecirclex2+y2+z2=9,x+y+z=1.
23. Findtheequationofthespherethroughthecirclex2+y2+z2=9,2x+3y+4z=5andthepoint(1,2,1).
Ans.:3(x2+y2+z2)–2x–2y–4z–22=0
24. Findtheequationofthespherecontainingthecirclex2+y2+z2–2x=9,z=0andthepoint(4,5,6).
Ans.:x2+y2+z2–2x–10z–9=0
25. Findtheequationofthespherepassingthroughthecirclex2+y2=a2,z=0andthepoint(α,β,λ).
Ans.:r(x2+y2+z2–a2)–z(α2+β2+γ2–a2)=0
26. Findtheequationofthespherethroughthecirclex2+y2+z2+2x+3y+6=0,x–2y+4z–9=0andthecentreofthesphere.
Ans.:x2+y2+z2–2x+4y–6z+5=0
27. Findtheequationsofthespherethroughthecirclex2+y2+z2=1,2x+4y+5z=6andtouchingtheplanez=0.
Ans.:x2+y2+z2–2x–4y–5z+5=0
5(x2+y2+z2)–2x–4y–5z+1=0
28. Findtheequationofthespherehavingthecirclex2+y2+z2+10y–4z–8=0,x+y+23=0asagreatcircle.
Ans.:x2+y2+z2–4x+6y–8z+4=0
29. Showthatthetwocirclesx2+y2+z2–y+2z=0,x–y+z–2=0andx2+y2+z2+x–3y+z–5=0,2x−y+4z−1=0lieonthesamesphereandfinditsequation.
Ans.:x2+y2+z2+3x–4y+5z–6=0
30. Provethatthecirclesx2+y2+z2–2x+3y+4z–5=0,5y+6z+1=0andx2+y2+z2–3x+4y+5z–6=0,x+2y–7z=0lieonthesamesphere.Finditsequation.
Ans.:x2+y2+z2–2x–2y–2z–6=0
31. Findtheconditionsthatthecirclesx2+y2+z2+2ux+2vy+2wz+d=0,lx+my+nz=pand
(x2+y2+z2)2u′x+2v′y+2w′z+d′=0,l′x+m′y+n′z=p′tolieonthesamecircle.
32. Findthecentreandradiusofthecircleformedbytheintersectionofthespherex2+y2+z2=2225andtheplane2x–2y+z=27.
Ans.:(6,–6,3),12
33. Findthecentreandradiusofthecirclex2+y2+z2=25,x+2y+2z=9.Ans.:(1,2,2),4
34. Findtheequationofthecirclewhichliesonthespherex2+y2+z2=25andhasthecentreat(1,2,3).
Ans.:x2+y2+z2=25,x+2y+3z=14
35. Aplanepassesthroughapoint(α,β,γ)andintersectsthespherex2+y2+z2=a2.Showthatthelocusofthecentreofthecircleofintersectionisthespherex(x–α)+y(y–β)+z(z–γ)=0.
36. Findtheequationofthespherethroughthecirclex2+y2+z2–4=0andthepoint(2,1,1).
Ans.:x2+y2+z2–2x+y–2z–1=0
37. Findtheequationofthespherethroughthecirclex2+y2+z2=9,2x+3y+4z=5andthroughtheorigin.
Ans.:5(x2+y2+z2)–18x–27y–36z=0
38. Showthatthetwocircles2(x2+y2+z2)+8x–13y+17z–17=0,2x+y–3z+1=0andx2+
y2+z2+3x–4y+3z=0,x–y+2z–4=0lieonthesamesphereandfinditsequation.
Ans.:x2+y2+z2+5x–6y–7z–8=0
39. Findtheequationofthespherewhichhasitscentreontheplane5x+y–4z+3=0andpassing
throughthecirclex2+y2+z2–3x+4y–2z+8=0,4x–5y+3z–3=0.
Ans.:x2+y2+z2+9x–11y+7z–1=0
40. FindtheequationofthespherewhichhasthecircleS=x2+y2+z2+2x+4y+6z–11=0,2x+
y+2z+1=0asgreatcircle.
Ans.:x2+y2+z2–2x+2y+2z–13=0.
41. Findtheequationofthespherewhoseradiusis1andwhichpassesthroughthecircleof
intersectionofthespheresx2+y2+z2+2x+2y+2z–6=0andx2+y2+z2+3x+3y–z–1=0.
Ans.:3x2+3y2+3z2+16x+16y+4z+32=0
42. Ifristheradiusofthecirclex2+y2+z2+2ux+2vy+2wz+d=0,lx+my+nz=0thenprove
that(r2+d2)(l2+m2+n2)=(mw–nv)2+(nv–lw)2+(lv–mu)2.
43. Findtheequationofthespherethroughthecirclex2+y2=4,z=0meetingtheplanex+2y+2z=0inacircleofradius3.
Ans.:x2+y2+z2–6z–4=0
44. Findtheequationofthespherethroughthecirclex2+y2+z2=1,2x+3y+4z=5andwhich
intersectthespherex2+y2+z2+3(x–y+z)–56=0orthogonally,x2+y2+z2–12x–18y–24z+29=0.
45. TheplaneABCwhoseequationis meetstheaxesinA,BandC.Findtheequationto
determinethecircumcircleofthetriangleABCandobtainthecoordinatesofitscentre.
46. Findtheequationofthecirclecircumscribingthetriangleformedbythethreepoints(a,0,0),(0,b,0),(0,0,c).Obtainthecoordinatesofthecentreofthecircle.
Ans.:x2+y2+z2–x–2y–3z=0,6x–3y–2z–6=0
Centre=
47. Findtheequationofthespherethroughthecirclex2+y2+z2+2x+3y+6=0,x–2y+4z–9=
0andthecentreofthespherex2+y2+z2–2x+4y–6z+5=0.
Ans.:x2+y2+z2+7y–8z+24=0
48. Findtheequationofthespherehavingitscentreontheplane4x–5y–z–3=0andpassing
throughthecirclex2+y2+z2–2x–3y+4z+8=0,x2+y2+z2+4x+5y–6z+2=0.
Ans.:x2+y2+z2+7x+9y–11z–1=0
49. Acirclewithcentre(2,3,0)andradiusunityisdrawnontheplanez=0.Findtheequationofthespherewhichpassesthroughthecircleandthepoint(1,1,1).
Ans.:x2+y2+z2–4x–6y–6z+12=0
50. Findtheequationofthespherewhichpassesthroughthecirclex2+y2=4,z=0andiscutbytheplanex+2y+2z=0inacircleofradius3.
Ans.:x2+y2+z2+6z–4=0,
x2+y2+z2–6z–4=0
51. Provethattheplanex+2y–z–4=0cutsthespherex2+y2+z2–x+z–2=0inacircleofradiusunityandalsofindtheequationofthespherewhichhasthiscircleasgreatcircle.
Ans.:x2+y2+z2–2x–2y+2z+2=0
52. Findtheequationofthespherehavingthecirclex2+y2+z2+10y–4z–3=0,x+y+z–3=0asagreatcircle.
Ans.:x2+y2+z2–4x+6y–8z+4=0
53. PisavariablepointonagivenlineandA,BandCareprojectionsontheaxes.ShowthatthesphereOABCpassesthroughafixedcircle.
Exercises3
1. Findtheequationsofthesphereswhichpassthroughthecirclex2+y2+z2=5,x+2y+3z=5andtouchtheplanez=0.
Ans.:x2+y2+z2–2x+y+5z+5=0
5(x2+y2+z2)–2x–4y+5z+1=0
2. Findtheequationsofthesphereswhichpassthroughthecirclex2+y2+z2–4x–y+3z+12=0,2x+3y–8z=10andtouchtheplanex–2y–2z=1.
Ans.:x2+y2+z2–2x+2y–4z+2=0
x2+y2+z2–6x–4y+10z+22=0
3. Showthatthetangentplaneat(1,2,3)tothespherex2+y2+z2+x+y+z–20=0is3x+5y+7z–34=0.
4. Findtheequationofthespherewhichtouchesthespherex2+y2+z2–x+3y+2z–3=0at(1,1,−1)andpassesthroughtheorigin.
Ans.:2x2+2y2+2z2–3x+y+4z=0
5. Findtheequationofthespherewhichtouchesthespherex2+y2+z2+2x–6y+1=0at(1,2,–2)andpassesthroughtheorigin.
Ans.:4(x2+y2+z2)+10x–28y–2z=0
6. Showthattheplane2x–2y+z+16=0touchesthespherex2+y2+z2+2x+4y+2z–3=0andfindthepointofcontact.
Ans.:(–3,4,–2)
7. Findtheequationofthetangentplaneattheorigintothespherex2+y2+z2–8x–6y+4z=0.Ans.:4x–3y+2z=0
8. Findtheequationofthetangentplanestothespherex2+y2+z2+2x–4y+6z–7=0whichpassesthroughtheline6x–3y–23=0,3z+2=0.
Ans.:8x−4y+z–34=0,4x–2y–z–16=0
9. Showthattheplane2x–2y+z–12=0touchesthespherex2+y2+z2–2x–4y+2z–3=0.
10. ShowthatthepointP(1,–3,1)liesonthespherex2+y2+z2+2x+2y–7=0andobtaintheequationofthetangentplaneatP.
Ans.:2x–2y+z=9
11. Ifthepoint(5,1,4)isoneextremitiesofadiameterofthespherex2+y2+z2–2x–2y–2z–22=0andfindthecoordinatesoftheotherextremity.Findtheequationtothetangentplanesatthetwoextremitiesandshowthattheyareparallel.
Ans.:(–3,1,–2);4x+3y–22=0,4x+3y+8=0
12. Findthevalueofkforwhichtheplanex+y+z=ktouchesthespherex2+y2+z2–2x–4y–6z+11=0.Findthepointofcontactineachcase.
Ans.:k=3or9;(0,1,2),(2,3,4)
13. Findtheequationtothetangentplanestothespherex2+y2+z2–2x–4y–6z–2=0whichareparalleltotheplanex+2y+2z–20=0.
Ans.:x+2y+2z–23=0x+2y+2z–1=0
14. Aspheretouchestheplanex–2y–2z–7=0atthepoint(3,–1,–1)andpassesthroughthepoint(1,1,–3).Findtheequation.
Ans.:x2+y2+z2–10y–10z–31=0
15. Showthattheline touchesthespherex2+y2+z2–6x+2y–4z+5=0.Find
thepointofcontact.Ans.:(5,–3,3)
16. Tangentplanesatanypointofthespherex2+y2+z2=r2meetsthecoordinateaxesatA,BandC.Showthatthelocusofthepointofintersectionoftheplanesdrawnparalleltothecoordinate
planesthroughthepointsA,BandCisthesurfacex–2+y–2+z–2=r–2.
17. Findtheconditionthattheline wherel,mandnarethedirectioncosinesofa
line,shouldtouchthespherex2+y2+z2+2ux+2vy+2wz+d=0.Showthattherearetwospheresthroughthepoints(0,0,0),(2a,0,0),(0,2b,0)and(0,0,2c)whichtouchtheaboveline
andthatthedistancebetweentheircentresis
18. Findtheequationofthespherewhichtouchesthespherex2+y2+z2+3y–x+2z–3=0at(1,1,–1)andpassesthroughtheorigin.
Ans.:2x2+2y2+2z2–3x+y+4z=0
19. Findtheequationofthetangentlineinsymmetricalformtothecirclex2+y2+z2+5x–7y+2z–8=0,3x–2y+4z+3=0.
20. Showthattheplane2x–2y–z+12=0touchesthespherex2+y2+z2–2x–4y+2z–3=0andfindthepointofcontact.
Ans.:(–1,4,–2)
21. Findtheequationofthespherewhichtouchesthespherex2+y2+z2+2x–6y+1=0atthepoint(1,2,–2)andpassesthroughtheorigin.
Ans.:4(x2+y2+z2)+10x–25y–22=0
22. Findtheequationsofthesphereswhichpassthroughthecirclex2+y2+z2=1,2x+4y+5z–6=0andtouchtheplanez=0.
Ans.:5(x2+y2+z2)–2x–4y–5z+6=0
5(x2+y2+z2)–2x–4y–5z+1=0
23. Findtheequationsofthespherepassingthroughthecirclex2+y2+z2–5=0,2x+3y+z–3=0andtouchingtheplane3x+4z−15=0.
Ans.:x2+y2+z2+4x+6y–2z–11=0
5(x2+y2+z2)–8x–12y–4z–37=0
24. Findthepointofintersectionofthelineandthespherex2+y2+z2–4x+6y–2z+5=0.Ans.:(4,–1,2),(0,–2,3)
25. Provethatthesumofthesquaresoftheinterceptsmadebyagivenlineonanythreemutuallyperpendicularlinesthroughafixedpointisconstant.
Exercises4
1. Provethatthespheresx2+y2+z2+6y+2z+8=0andx2+y2+z2+6x+8y+4z+2=0intersectorthogonally.
2. Findtheequationofthespherewhichpassesthroughthecirclex2+y2+z2–2x+3y–4z+6=0,
3x–4y+5z–15=0andwhichcutsorthogonallythespherex2+y2+z2+2x+4y–6z+11=0.
Ans.:5(x2+y2+z2)–13x+19y–25z+45=0
3. Findtheequationofthespherethatpassesthroughthecirclex2+y2+z2–2x+3y–4z+6=0,
3x–4y+5z–15=0andwhichcutsthespherex2+y2+z2+2x+4y+6z+11=0orthogonally.
Ans.:x2+y2+z2+x–y+z–9=0
4. Provethateveryspherethroughthecirclex2+y2–2ax+r2=0,z=0cutsorthogonallyevery
spherethroughthecirclex2+z2=r2,y=0.5. Findtheequationofthespherewhichtouchestheplane3x+2y–z+7=0atthepoint(1,−2,1)
andcutsthespherex2+y2+z2–4x+6y+4=0orthogonally.
Ans.:3(x2+y2+z2)+6x+20y–10z+36=0
6. Findtheequationofthespherethatpassesthroughthepoints(a,b,c)and(–2,1,–4)andcuts
orthogonallythetwospheresx2+y2+z2+x–3y+2=0and(x2+y2+z2)+x+3y+4=0.
Ans.:x2+y2+z2+2x–2y+4z–3=0
7. Findtheequationofthespherewhichtouchestheplane3x+2y–z+2=0atthepointP(1,–2,1)andalsocutsorthogonally.
Ans.:x2+y2+z2+7x+10y–5z+12=0
8. Ifdisthedistancebetweenthecentresofthetwospheresofradiir1andr2thenprovethatthe
anglebetweenthemis
9. Findtheconditionthatthespherea(x2+y2+z2)+2lx+z–y+2nz+p=0andb(x2+y2+z2)k2
maycutorthogonally.
Ans.:ak2=bp2
10. Findtheequationoftheradicalplanesofthespheresx2+y2+z2+2x+2y+2z–2=0,x2+y2+
z2+4y=0,x2+y2+z2+3x–2y+8z–6=0.Ans.:x–y+z–1=0,3x–6y+8z–6=0
x–4y+6z+4=0
11. Findtheequationoftheradicallineofthespheres(x–2)2+y2+z2=1,x2+(y–3)2+z2=6
and(x+2)2+(y+1)2+(z–2)2=6.
12. Findtheequationoftheradicallineofthespheresx2+y2+z2+2x+2y+2z+2=0,x2+y2+
z2+4y=0,x2+y2+z2+3x–2y+8z+6=0.Ans.:x–y+z+1=0,3x–6y+8z+6=0
13. Findtheradicalplaneofthespheresx2+y2+z2–8x+4y+4z+12=0,x2+y2+z2–6x+3y+3z+9=0.
Ans.:2x–y–z–3=0
14. Findthespherescoaxalwiththespheresx2+y2+z2+2x+y+3z–8=0andx2+y2+z2–5=0andtouchingtheplane3x+4y=15.
Ans.:5(x2+y2+z2)–8x–4y–12z–13=0
15. Findthelimitingpointsofthecoaxalsystemdefinedbythespheresx2+y2+z2+3x–3y+6=0,
x2+y2+z2–6y–6z+6=0.Ans.:(–1,2,1),(–2,1,–1)
16. Findthelimitingpointsofthecoaxalsystemdeterminedbythetwosphereswhoseequationsare
x2+y2+z2–8x+2y–2z+32=0,x2+y2+z2–7x+z+23=0.Ans.:(3,1,–2),(5,–3,4)
17. Findtheequationsofthesphereswhoselimitingpointsare(–1,2,1)and(–2,1,–1)andwhichtouchestheplane2x+3y+6z+7=0.
18. Findtheequationofthespherewhichtouchestheplane3x+2y–z+2=0atthepoint(1,–2,1)
andalsocutorthogonallythespherex2+y2+z2–4x+6y+4=0.
Ans.:x2+y2+z2+7x+10y–5z+12=0
19. Findthelimitingpointsofthecoaxalsystemtwoofwhosemembersarex2+y2+z2–3x–3y+6
=0,x2+y2+z2–4y–6z+6=0.Ans.:(2,–3,4)and(–2,3,–4)
20. Thepoint(–1,2,1)isalimitingpointofacoaxalsystemofspheresofwhichthespherex2+y2+
z2+3x–2y+6=0isamember.Findthecoordinatesoftheotherlimitingpoint.Ans.:(–2,1,–1)
Chapter15
Cone
15.1DEFINITIONOFCONE
Aconeisasurfacegeneratedbyastraightlinewhichpassesthroughafixedpointandintersectsafixedcurveortouchesagivencurve.Thefixedpointiscalledthevertexoftheconeandthefixedcurveiscalleda
guidingcurveofthecone.Thestraightlineiscalledagenerator.
15.2EQUATIONOFACONEWITHAGIVENVERTEXANDAGIVENGUIDINGCURVE
Let(α,β,γ)bethegivenvertexandax2+2hxy+by2+2gx+2fy+c=0,z=0betheguidingcurve.
Theequationsofanylinepassingthroughthepoint(α,β,γ)are
Whenthislinemeetstheplaneatz=0weget,
Thispointliesonthegivencurveax2+2hxy+by2+2gx+2fy+c=0,z=0.
Eliminatingl,m,nfrom(15.1)and(15.2)wegettheequationofthecone.
From(15.1)
Multiplyingthroughoutby(z−γ)2,weget
a(αz−γx)2+2h(αz−γx)(βz−γy)+b(βz−γy)2+2g(αz−γx)(z−γ)+zf(βz−γy)(z−γ)+c(z−γ)2=0.
Thisistherequiredequationofthecone.
Example15.2.1
Findtheequationoftheconewithitsvertexat(1,1,1)andwhichpassesthroughthecurvex2+y2=4,z=2.
Solution
LetVbethevertexoftheconeandPbeanypointonthesurfaceofthecone.LettheequationsofthegeneratorVPbe
Thislineintersectstheplanez=2.
Thispointliesonthecurvex2+y2=4.
Eliminatingl,m,nfrom(15.3)and(15.4)weget
or
Thisisrequiredequationofthecone.
Example15.2.2
Findtheequationoftheconewhosevertexis(a,b,c)andwhosebaseisthecurveαx2+βy2=1,z=0.
Solution
ThevertexoftheconeisV(a,b,c).
Theguidingcurveis
Letl,m,nbethedirectioncosinesofthegeneratorVP.
ThentheequationsofVPare
Whenthislinemeetsz=0wehave
Thepointliesonthecurveαx2+βy2=1.
Wehavetoeliminatel,m,nfrom(15.6)and(15.7)
Thisistheequationoftherequiredcone.
15.3EQUATIONOFACONEWITHITSVERTEXATTHEORIGIN
Toshowthattheequationofaconewithitsvertexattheoriginishomogeneous,let
betheequationofaconewithitsvertexattheorigin.LetP(α,β,γ)beanypointonthesurface.Then,OPisageneratorofthecone.
Since(α,β,γ)liesonthecone
TheequationsofOPare
Anypointonthislineis(tα,tβ,tγ).Thepointliesontheconef(x,y,z)=0.
Fromequations(15.8)and(15.11),weobservethattheequationf(x,y,z)=0ishomogeneous.
Conversely,everyhomogeneousequationin(x,y,z)representsaconewithitsvertexattheorigin.
Letf(x,y,z)=0beahomogeneousequationinx,y,z.Sincef(x,y,z)=0isahomogeneousequation,f(x,y,z)=0foranyreal
number.Inparticularf(0,0,0)=0.
Therefore,theoriginliesonthelocusoftheequation(15.8).Asf(tx,ty,tz)=0,anypointonthelinethroughtheoriginliesontheequation(15.8).Inotherwords,thelocusof(15.8)isasurfacegeneratedbythelinethroughtheorigin.Henceequationrepresentsaconewithitsvertexattheorigin.
Note15.3.1:Iff(x,y,z)canbeexpressedastheproductofnlinearfactorsthenf(x,y,z)=0representsnplanesthroughtheorigin.
Note15.3.2:Iff(x,y,z)=0isahomogeneousequationofseconddegreeinx,yandzthenf(x,y,z)=0isaquadriccone.Ifitcanbefactoredintotwolinearfactorsthenitrepresentsapairofplanesthroughtheorigin;thenweregardequationf(x,y,z)=0asdegeneratecone,thevertexbeinganypointonthelineofintersectionofthetwoplanes.
Generators:Theline isageneratoroftheconef(x,y,z)=0withits
vertexattheoriginifandonlyiff(l,m,n)=0.
Let beageneratoroftheconef(x,y,z)=0thenthepoint(lr,
mr,nr)liesonthecone.Takingr=1,thepoint(l,m,n)liesontheconef(x,y,z)=0.
∴f(l,m,n)=0
Converse:Letf(x,y,z)=0betheequationoftheconewithitsvertexatthe
originand bealinethroughtheoriginsuchthatf(l,m,n)=0.
Sincethevertexisattheorigin,f(x,y,z)=0isahomogeneousequationinx,yandz.
Nowwewillprovethat isageneratorofthecone.
Anypointonthegeneratoris(lr,mr,nr).Sincef(x,y,z)=0andf(l,m,n)=0,itfollowsthatf(lr,mr,nr)=0.
∴ isageneratoroftheconef(x,y,z)=0.
Example15.3.1
Findtheequationoftheconewithitsvertexattheoriginandwhichpassesthroughthecurveax2+by2+cz2−1=0=αx2+βy2−2z.
Solution
Lettheequationofthegeneratorbe
Anypointonthislineis(lr,mr,nr).Thispointliesonthecurve
From(15.14),
Substitutingthisin(15.13)weget
Asl,m,nareproportionaltox,y,ztheequationoftheconeis4z2(ax2+by2+cz2)=(αx2+βy2)2.
Example15.3.2
Findtheequationoftheconewhosevertexisattheoriginandtheguidingcurve
is
Solution
Sincethevertexoftheconeistheoriginitsequationmustbeahomogeneousequationofseconddegree.Theequationsoftheguidingcurveare
Homogenizingtheequation(15.15)withthehelpof(15.16)wegettheequation
oftherequiredcone.Hencetheequationoftheconeis
(i.e.)27x2+32y2+7z2(xy+yz+zx)=0
Example15.3.3
Theplane meetsthecoordinateaxesatA,BandC.Provethatthe
equationoftheconegeneratedbylinesdrawnfromOtomeetthecircleABCis
Solution
ThepointsA,B,Care(a,0,0),(0,b,0)and(0,0,c),respectively.TheequationofthesphereOABCisx2+y2+z2−ax−by−cz=0.TheequationsofthecircleABCare
Homogenizingequation(15.17)withthehelpof(15.18)wegettheequationoftherequiredcone.
15.4CONDITIONFORTHEGENERALEQUATIONOFTHESECONDDEGREETOREPRESENTACONE
Letthegeneralequationoftheseconddegreebe
Let(x1,y1,z1)bethevertexofthecone.Shifttheorigintothepoint(x1,y1,z1).Then
Thentheequation(15.19)becomes
SincethisequationhastobeahomogeneousequationinX,YandZ.CoefficientofX=0CoefficientofY=0CoefficientofZ=0andconstantterm=0.
Eliminatingx,y,zfrom(15.21),(15.22),(15.23)and(15.24),weget,
Thisistherequiredcondition.
Note15.4.1:Ifthegivenequationoftheseconddegreeisf(x,y,z)=0thenmakeithomogeneousbyintroducingthevariabletwheret=1.Then
Solvinganythreeofthesefourequations,wegetthevertexofthecone.Testwhetherthesevaluesofx,y,zsatisfythefourthequation.
Example15.4.1
Findtheequationoftheconeoftheseconddegreewhichpassesthroughtheaxes.
Solution
Theconepassesthroughtheaxes.Therefore,thevertexoftheconeistheorigin.Theequationsoftheconeisahomogeneousequationofseconddegreeinx,y
andz.
Giventhatx-axisisagenerator.Theny=0,z=0mustsatisfytheequation(15.25)
∴a=0Sincey-axisisageneratorb=0.Sincez-axisisageneratorc=0.
Hencetheequationoftheconeisfyz+gzx+hxy=0.
Example15.4.2
Showthatthelinesthroughthepoint(α,β,γ)whosedirectioncosinessatisfytherelational2+bm2+cn2=0,generatestheconea(x−α)2+b(y−β)2+c(z−γ)2
=0.
Solution
Theequationsofanylinethrough(α,β,γ)are
where
Eliminatingl,m,nfrom(15.26)and(15.27)
weget,
a(x−α)2+b(y−β)2+c(z−γ)2=0Example15.4.3
Findtheequationtothequadricconewhichpassesthroughthethreecoordinate
axesandthethreemutuallyperpendicularlines ,
Solution
Wehaveseenthattheequationoftheconepassingthroughtheaxesis
Thisconepassesthroughline
or
Sincetheconealsopassesthroughtheline wehave
From(15.29)and(15.30)weget
From(15.28)and(15.31)weget5yz+8zx−3xy=0.
Sincetheconepassesthroughtheline
weget4f+5g+20h=0and4(5)+5(8)+20(−3)=0isalsotrue.Therefore,theequationoftherequiredconeis5yz+8zx−3xy=0.
Example15.4.4
Provethattheequation2x2+2y2+7z2−10yz−10zx+2x+2y+26z−17=0representsaconewhosevertexis(2,2,1).
Solution
LetF(x,y,z,t)=2x2+2y2+7z2−10yz−10zx−2xt+2yt+26zt−17t2=0
givetheequations
Solvingthefirstthreeequationswegetx=2,y=2,z=1.Thesevaluesalsosatisfythefourthequation.Therefore,thegivenequationrepresentsaconewithitsvertexatthepoint(2,2,1).
Exercises1
1. Findtheequationoftheconewhosevertexisattheoriginandwhichpassesthroughthecurveof
intersectionoftheplanelx+my+nz=pandthesurface,ax2+by2+cz2=1.
Ans.:p2(ax2+by2+cz2)=(lx+my+nz)2
2. Findtheequationoftheconewhosevertexis(α,β,γ)andwhoseguidingcurveistheparabolay2
=4ax,z=0.
Ans.:(ry−βz)2=4a(z−γ)(αz−rx)
3. Provethattheline where2l2+3m2−5n2=0isageneratorofthecone2x2+3y2−5z2
=0.
4. Findtheequationoftheconewhosevertexisatthepoint(1,1,0)andwhoseguidingcurveisx2+
z2=4,y=0.
Ans.:x2−3y2+z2−2xy+8y−4=0
5. Findtheequationoftheconewhosevertexisthepoint(0,0,1)andwhoseguidingcurveisthe
ellipse ,z=3.Alsoobtainsectionoftheconebytheplaney=0andidentifyitstype.
Ans.:36x2+100y2−225z2+450z−225=0
pairofstraightlines
6. Findtheequationsoftheconeswithvertexattheoriginandpassingthroughthecurvesof
intersectiongivenbytheequations:
i.
ii.
iii.
iv.
v.
7. Theplanex+y+z=1meetsthecoordinateaxesinA,BandC.Provethattheequationtothe
conegeneratedbythelinesthroughO,tomeetthecircleABCisyz+zx+xy=0.
8. Avariableplaneisparalleltotheplane andmeetstheaxesinA,BandC.Provethat,
thecircleABCliesonthecone
9.i. Findtheequationofthequadricconewhichpassesthroughthethreecoordinateaxesand
threemutuallyperpendicularlines
Ans.:16yz−33zx−25xy=0
ii. Provethattheequationoftheconewhosevertexis(0,0,0)andthebasecurvez=k,f(x,
y)=0is wheref(x,y)=ax2+2hxy+by2+2gx+2fy+c=0.
10. Findtheequationtotheconewhosevertexistheoriginandthebasecirclex=a,y2+z2=b2andshowthatthesectionoftheconebyaplaneparalleltothexy-planeishyperbola.
Ans.:a2(y2+z2)=b2x2
11. PlanesthroughOXandOYincludeanangleα.Showthatthelineofintersectionliesonthecone
z2(x2+y2+z2)=x2y2tan2α.12. Provethataconeofseconddegreecanbefoundtopassthroughtwosetsofrectangularaxes
throughthesameorigin.
13. Provethattheequationx2−2y2+3z2+5yz−6zx−4xy+8x−19y−2z−20=0representsaconewithitsvertexat(1,−2,3).
14. Provethattheequation2y2−8yz−4zx−8xy+6x−4y−2z+5=0representsaconewhose
vertexis .
15. Provethattheequationax2+by2+cz2+2ux+2vy+2wz+d=0representsaconeif
15.5RIGHTCIRCULARCONE
Arightcircularconeisasurfacegeneratedbyastraightlinewhichpassesthroughafixedpoint,andmakesaconstantanglewithafixedstraightline
throughthefixedpoint.Thefixedpointiscalledthevertexoftheconeandtheconstantangleiscalledthesemiverticalangleandfixedstraightlineiscalledtheaxisofthecone.Thesectionofrightcircularconebyanyplaneperpendiculartoitsaxisisa
circle.
15.5.1EquationofaRightCircularConewithVertexV(α,β,γ),AxisVLwithDirectionRatiosl,m,nandSemiverticalAngleθ
LetP(x,y,z)beanypointonthesurfaceofthecone.ThendirectionratiosofVParex−α,y−β,z−γ.
ThedirectionratiosoftheperpendicularVLarel,m,n.
Thisistherequiredequationofthecone.
Note15.5.1.1:
i. Ifthevertexisattheoriginthentheequationoftheconebecomes(lx+my+nz)2=[(x2+y2+
z2)(l2+m2+n2)]cos2θ.ii. Ifl,m,narethedirectioncosinesofthelinethen
iii. Ifaxisofconeisthez-axisthentheequation(15.33)becomes
or
15.5.2EnvelopingCone
Ithasbeenseeninthetwo-dimensionalanalyticalgeometrythattwotangentscanbedrawnfromagivenpointtoaconic.Inanalogywiththataninfinitenumberoftangentlinescanbedrawnfromagivenpointtoaconicoid,inparticulartoasphere.Allsuchtangentlinesgenerateaconewiththegivenpointasvertex.Suchaconeiscalledanenvelopingcone.
Definition15.5.2.1:Thelocusoftangentlinesdrawnfromagivenpointtoagivensurfaceiscalledanenvelopingconeofthesurface.Thegivenpointiscalledthevertexofthecone.
Equationofenvelopingcone:Letusfindtheequationoftheenvelopingconeofthespherex2+y2+z2=a2withthevertexat(x,y,z).
LetP(x,y,z)beanypointonthetangentdrawnfromV(x1,y1,z1)tothegivensphere.LetQbethepointthatdividesPQintheratio1:λ.Thenthecoordinates
ofQare
Ifthispointliesonthespherethen,
Thisisaquadraticequationinλ.TherearetwovaluesofλindicatingthattherearetwopointsonVPwhichdividesPQintheratio1:λandlieonthesphere.IfPQisatangenttothespherethenthesetwopointscoincideandthepointisthepointofcontact.Therefore,thetwovaluesofλofequation(15.34)mustbeequalandhencethe
discriminantmustbezero.
Thisistheequationofthe
envelopingcone.
Note15.5.2.2:Theequationoftheenvelopingconecanbeexpressedintheform
Example15.5.1
Findtheequationoftherightcircularconewhosevertexisattheorigin,whose
axisistheline andwhichhasaverticalangleof60°.
Solution
Theaxisoftheconeis .
Therefore,thedirectionratiosoftheaxisoftheconeare1,2,3.
Thedirectioncosinesare .
LetP(x,y,z)beanypointonthesurfaceofthecone.LetPLbeperpendiculartoOA.
Example15.5.2
Findtheequationoftherightcircularconewithitsvertexattheorigin,axisalongthez-axisandsemiverticalangleα.
Solution
Thedirectioncosinesoftheaxisoftheconeare0,0,1.
LetP(x,y,z)beanypointonthecone.
Then,
Thisistherequiredequationofthecone.
Example15.5.3
Findthesemiverticalangleandtheequationoftherightcircularconehavingitsvertexatoriginandpassingthroughthecircley2+z2=b2,x=a.
Solution
Theguidingcircleoftherightcircularconeisy2+z2=b2,x=a.Therefore,theaxisoftheconeisalongx-axis.
Ifθisthesemiverticalangle,then .
LetP(x,y,z)beanypointonthesurfaceofthecone.ThedirectionratiosofOParex,y,z.
Thedirectioncosinesofthex-axisare1,0,0.
whichistherequiredequationofthecone.
Example15.5.4
Arightcircularconehasitsvertexat(2,−3,5).ItsaxispassesthroughA(3,−2,6)anditssemiverticalangleis30°.Finditsequation.
Solution
Theaxisisthelinejoiningthepoints(2,−3,5)and(3,−2,6).
Therefore,itsequationsare .
Thedirectionratiosoftheaxesare1,1,1.
Thedirectionratiosoftheaxesare1,1,1.
Thedirectioncosinesare
LetP(x,y,z)beanypointonthecone.
Squaring,crossmultiplyingandsimplifyingweget,5x2+5y2+5z2−8xy−8yz−8zx−4x+86y−58z+278=0.
Example15.5.5
Arightcircularconehasthreemutuallyperpendiculargenerators.Provethatthe
semiverticalangleoftheconeis
Solution
Theequationoftherightcircularconewithvertexattheorigin,semiverticalangleαandaxisalongz-axisisgivenbyx2+y2=z2tan2α.Thisconewillhavethreemutuallyperpendiculargeneratorsifcoefficientof
x2+coefficientofy2+coefficientofz2=0.
Example15.5.6
TheaxisofarightconevertexO,makesequalangleswiththecoordinateaxesandtheconepassesthroughthelinedrawnfromOwithdirectioncosinesproportionalto(1,−2,2).Findtheequationtothecone.
Solution
Lettheaxisoftheconemakeanangleβwiththeaxes.Thenthedirectioncosinesoftheaxesarecosβ,cosβ,cosβ.(i.e.)1,1,1.Letαbethesemiverticalangleoftheaxisofthecone.Thedirectionratiosofoneofthegeneratorsare1,−2,2.
LetP(x,y,z)beanypointonthecone.ThenthedirectionratiosofOParex,y,z.Thedirectionratiosoftheaxisare1,1,1.
Squaringandcrossmultiplyingweget,
9(x+y+z)2=x2+y2+z2or4x2+4y2+4z2+9xy+9yz+9zx=0
Example15.5.7
Provethatthelinedrawnfromtheoriginsoastotouchthespherex2+y2+z2+2ux+2vy+2wz+d=0lieontheconed(x2+y2+z2)=(ux+vy+wz)2.
Solution
Thelinesdrawnfromtheorigintotouchthespheregeneratestheenvelopingcone.
TheequationoftheenvelopingconeofthegivensphereisT2=SS1.
Example15.5.8
Showthattheplanez=0cutstheenvelopingconeofthespherex2+y2+z2=11whichhasitsvertexat(2,4,1)inarectangularhyperbola.
Solution
Theequationoftheenvelopingconewithitsvertexat(2,4,1)isT2=SS1.
Thesectionofthisconebytheplanez=0is(2x+4y−11)2=10(x2+y2−11).Coefficientofx2+coefficientofy2=6−6=0Hence,theplanez=0cutstheenvelopingconeinarectangularhyperbola.
Exercises2
1. Findtheequationoftherightcircularconewhosevertexistheline andwhichhasa
verticalangleof60°.
Ans.:19x2+13y2+3z2−8xy−24yz−12zx=0
2. If(x,y,z)isanypointontheconewhosevertexis(1,0,2)andsemiverticalangleis30°andthe
equationtotheaxisis ,showthattheequationoftheconeis27[(x−1)2+y2+(z−
2)2]=4(x+2y−2z+3)2.3. Findtheequationtotherightcircularconeofsemiverticalangle30°,whosevertexis(1,2,3)and
whoseaxisisparalleltothelinex=y=z.
Ans.:5(x2+y2+z2)−8(yz+zx+xy)+30x+12y−6z−18=0
4. Findtheequationtotherightcircularconewhosevertexis(3,2,1),semiverticalangleis30°and
axisistheline
Ans.:7x2+37y2+21z2−16xy−12yz−48zx+38x−88y+126z−32=0
5. Findtheequationoftherightcircularconewithvertexat(1,−2,−1),semiverticalangle60°and
axis
Ans.:5[(5x+4y+14)2+(3z−5x+8)2+(4x+3y+2)2]
=75[(x−1)2+(y+2)2+(z+1)2]
6. Findtheequationoftherightcircularconewhichpassesthroughthethreelinesdrawnfromthe
originwithdirectionratios(1,2,2)(2,1,−2)(2,−2,1).
Ans.:8x2−5y2−4z2+yz+5zx+5xy=0
7. Linesaredrawnthroughtheoriginwithdirectioncosinesproportionalto(1,2,2),(2,3,6),(3,4,12).Findtheequationoftherightcircularconethroughthem.Alsofindthesemiverticalangleofthecone.
Ans:
8. Findtheequationoftheconegeneratedwhenthestraightline2y+3z=6,x=0revolvesaboutthez-axis.
Ans.:4x2+4y2−9z2+36z−36=0
9. Findtheequationtotherightcircularconewhichhasthethreecoordinateaxesasgenerators.Ans.:xy+yz+zx=0
10. Findtheequationoftherightcircularconewithitsvertexatthepoint(0,0,0),itsaxisalongthey-axisandsemiverticalangleθ.
Ans.:x2+z2=y2tan2θ
11. Ifαisthesemiverticalangleoftherightcircularconewhichpassesthroughthelinesox,oy,x=y
=z,showthat
12. Provethatx2+y2+z2−2x+4y+6z+6=0representsarightcircularconewhosevertexisthepoint(1,2,−3),whoseaxisisparalleltooyandwhosesemiverticalangleis45°.
13. Provethatthesemiverticalangleofarightcircularconewhichhasthreemutuallyperpendicular
tangentplanesis
14. Findtheenvelopingconeofthespherex2+y2+z2−2x−2y=2withitsvertexat(1,1,1).
Ans.:3x2−y2+4zx−10x+2y−4z+6=0
15. Findtheenvelopingconeofthespherex2+y2+z2=11whichhasitsvertexat(2,4,1)andshowthattheplanez=0cutstheenvelopingconeinarectangularhyperbola.
15.6TANGENTPLANE
Tangentplanefromthepoint(x1,y1,z1)totheconeax2+by2+cz2+2fyz+2gzx+2hxy=0.Theequationsofanylinethroughthepoint(x1,y1,z1)are
Anypointonthelineis(x1+lr,y1+mr,z1+nr).
Ifthispointliesonthegivenconethen
Thisequationisaquadraticinr.Since(x1,y1,z1)isapointonthecone
Therefore,onerootoftheequation(15.36)iszero.Ifline(15.35)isatangenttothecurve,thenboththerootsareequalandhencetheotherrootmustbezero.Theconditionforthisisthecoefficientofr=0.
Hencethisistheconditionfortheline(15.35)tobeatangenttothecurveatthepoint(x1,y1,z1).Sinceequation(15.38)canbesatisfiedforinfinitelymanyvaluesofl,m,n
thereareinfinitelymanytangentlinesatanypointofthecone.Thelocusofallsuchtangentlinesisobtainedbyeliminatingl,m,nfrom
(15.35)and(15.38).
Therefore,theequationofthetangentplaneat(x1,y1,z1)is
Note15.6.1:(0,0,0)satisfiestheaboveequationandhencethetangentplaneatanypointofaconepassesthroughthevertex.
ThetangentplaneatanypointofaconetouchestheconealongthegeneratorthroughP.
Proof:Lettheequationoftheconebeax2+by2+cz2+2fyz+2gzx+2hxy=0.LetP(x1,y1,z1)beanypointonthecone.TheequationofthetangentplaneatPis(ax1+hy1+gz1)x+(hx1+by1+
fz1)y+(gx1+fy1+cz1)z=0.
TheequationsofthegeneratorthroughPare
Anypointonthislineis(rx1,ry1,rz1).Theequationofthetangentplaneat(rx1,ry1,rz1)is(arx1+hry1+grz1)x+(hrx1+bry1+frz1)y+(grx1+fry1+crz1)z=0.
Dividingbyr,(ax1+hy1+gz1)x+(hx1+by1+fz1)y+(gx1+fy1+cz1)z=0whichisalsotheequationofthetangentplaneat(x1,y1,z1).Therefore,thetangentplaneatPtouchestheconealongthegeneratorthroughP.
15.6.1ConditionfortheTangencyofaPlaneandaCone
Lettheequationoftheconebe
Lettheequationoftheplanebe
Lettheplane(15.40)touchtheconeat(x1,y1,z1).Theequationofthetangentplaneat(x1,y1,z1)is
Iftheplane(15.40)touchesthecone(15.39)thenequations(15.40)and(15.41)areidentical.Thereforeidentifying(15.40)and(15.41)weget,
Alsoas(x1,y1,z1)liesontheplane
Eliminating(x1,y1,z1),r1from(15.43),(15.44),(15.45)and(15.46)weget,
Simplifyingthisweget,
whereA,b,C,F,G,Harethecofactorsofa,b,c,f,g,hinthedeterminant
Hence(15.47)istherequiredconditionfortheplane(15.40)totouchthecone.
15.7RECIPROCALCONE
15.7.1EquationoftheReciprocalCone
Letusnowfindtheequationoftheconereciprocaltothecone
Letatangentplanetotheconebe
Thenwehavethecondition
whereA,b,C,F,G,Harethecofactorsofa,b,c,f,g,hinthedeterminant
Theequationofthelinethroughthevertex(0,0,0)ofthecone(15.48)andnormaltothetangentplane(15.49)are
Thelocusof(4)whichisgotbyeliminatingl,m,nfrom(15.50)and(15.51)is
Thisistheequationofthereciprocalcone.
Note15.7.1.1:Ifwefindthereciprocalconeof(15.52)wegettheequationofconeasax2+by2+cz2+2fyz+2gzx+2hxy=0.
Definition15.7.1.2:Twoconesaresaidtobereciprocalconesofeachotherifeachoneisthelocusofthenormalthroughthevertextothetangentplanesoftheother.
15.7.2AnglebetweenTwoGeneratingLinesinWhichaPlaneCutsaCone
Lettheequationoftheconeandtheplanebe
Letoneofthelinesofthesectionbe
Sincethislineliesontheplanewehave,
from(15.56)
Substitutingthisin(15.55)weget,
Thisisaquadraticequationin
Therearetwovaluesfor
Hencethegivenplaneintersectstheconeintwolinesnamely,
Alsosince aretherootsoftheequation(15.57)
whereP2=−(Au2+Bv2+Cw2+2Fvw+2Gwu+2Huv)andA,B,C,F,G,H
arethecofactorsofa,b,c,fg,hinthedeterminant
Itfollowsbysymmetry
Ifθistheacuteanglebetweenthelinesthen
Note15.7.2.1:Ifthetwolinesareperpendicularthen(a+b+c)(u2+v2+w2)−f(u,v,w)=0.
(i.e.)f(u,v,w)=(a+b+c)(u2+v2+w2)Note15.7.2.2:Ifthelinesofintersectionarecoincidentthenθ=0.
Thisistheconditionfortheplaneux+vy+wz=0tobeatangentplanetothecone.
15.7.3ConditionforMutuallyPerpendicularGeneratorsoftheCone
Wehaveseenthattheconditionfortheplaneux+vy+wz=0cuttheconeintwoperpendiculargeneratorsisthat
Ifthereisathirdgeneratorwhichisperpendiculartotheabovetwolinesofintersectionthenitmustbeanormaltotheplaneux+vy+wz=0.Therefore,itsequationsare
Since(15.59)isageneratoroftheconef(x,y,z)=0,weget
(15.58)and(15.60)holdsifandonlyifa+b+c=0.Therefore,theconditionfortheconetohavethreemutuallyperpendicular
generatorsisa+b+c=0.
Example15.7.1
Findtheanglebetweenthelinesofsectionoftheplane3x+y+5z=0andthecone6yz−2zx+5xy=0.
Solution
Lettheequationsofthelineofsectionbe
Asthislineliesontheplaneandalsoontheconeweget
From(15.62)m=−(3l+5n)Substitutingthisin(15.63)weget,
Solvingl+n=0and3l+m+5n=0weget,
Solvingl+2n=0and3l+m+5n=0weget,
Therefore,thedirectionratiosofthetwolinesare1,2,−1and2,−1,−1.
Ifθistheanglebetweenthelines
Therefore,theacuteanglebetweenthelinesis
Example15.7.2
Provethattheanglebetweenthelinesgivenbyx+y+z=0,ayz+bzx+cxy=
0is ifa+b+c=0.
Solution
Theplane
willcutthecone
intwolinesthroughthevertex(0,0,0).Theequationsofthelinesofthesectionare
wherel,m,narethedirectionratiosofthelines.Sincethelinegivenby(15.66)liesontheplaneandthecone
Substitutingn=−(l+m)in(15.67)weget
Ifl1,m1,n1andl2,m2,n2arethedirectionratiosofthetwolinesweget,
Similarlywecanshowthat
Ifθistheanglebetweenthelines
Example15.7.3
Provethattheconesax2+by2+cz2=0and arereciprocal.
Solution
Theequationofthereciprocalconeax2+by2+cz2=0is
whereA,B,Carethecofactorsofa,b,cin
Theequationofthereciprocalconeisbcx2+cay2+abz2=0or .
Similarly,wecanshowthatthereciprocalconeof isax2+by2+
cz2=0.Therefore,thetwogivenconesarereciprocaltoeachother.
Example15.7.4
Showthattheequation representsaconewhichtouchesthe
coordinateplanes.
Solution
Squaring(fx+gy−hz)2=4fgxy
Thisbeingahomogeneousequationofseconddegreeinx,y,z,itrepresentsacone.Whenthisconemeetstheplanex=0weget,(gy−hz)2=0.Hencetheaboveconeiscutbytheplanex=0incoincidentlinesandhencex
=0touchesthecone.Similarly,y=0,z=0alsotouchthecone.
Exercises3
1. Findtheanglebetweenthelinesofthesectionoftheplanesandcones:
i. x+3y−2z=0,x2+9y2−4z2=0
ii. 6x−10y−7z=0,108x2−20y2−7z2=0
Ans:
2. Showthattheanglebetweenthelinesinwhichtheplanex+y+z=0cutstheconeayz+bzx+
cxy=0is
3. Provethattheequationa2x2+b2y2+c2z2−2bcyz−2cazx−2abxy=0representsaconewhichtouchesthecoordinateplane.
4. If representsoneofthegeneratorsofthethreemutuallyperpendiculargeneratorsofthe
cone5yz−8zx−3xy=0thenfindtheothertwo.
Ans:
5. If representsoneofthethreemutuallyperpendiculargeneratorsofthecone11yz+6zx−
14xy=0thenfindtheothertwo.
Ans:
Chapter16
Cylinder
16.1DEFINITION
Thesurfacegeneratedbyavariablelinewhichremainsparalleltoafixedlineandintersectsagivencurve(ortouchesagivensurface)iscalledacylinder.Thevariablelineiscalledthegenerator,thefixedstraightlineiscalledthe
axisofthecylinderandthegivencurveiscalledtheguidingcurveofthecylinder.
16.2EQUATIONOFACYLINDERWITHAGIVENGENERATORANDAGIVENGUIDINGCURVE
Letusfindtheequationofthecylinderwhosegeneratorsareparalleltotheline
andwhoseguidingcurveistheconic
Let(α,β,γ)beanypointonthecylinder.Thentheequationsofageneratorare
Letusfindthepointwherethislinemeetstheplanez=0.Whenz=0,
Thispointis
Whenthegeneratormeetstheconic,thispointliesontheconic.
Thelocusofthepoint(α,β,γ)is
Thisistherequiredequationofthecylinder.
Note1:Ifthegeneratorsareparalleltoz-axisl=0,m=0,n=lthentheequationofthecylinderbecomes,
Note2:Theequationf(x,y)=0inspacerepresentsacylinderwhosegeneratorsareparalleltoz-axis.
16.3ENVELOPINGCLINDER
Thelocusofthetangentlinesdrawntoasphereandparalleltoagivenline
Letthegivenspherebe
Letthegivenlinebe
Let(α,β,γ)beanypointonthelocus.Thenanylinethrough(α,β,γ)paralleltoline(16.5)is
Anypointonthislineis(α+lr,β+mr,γ+nr)Ifthepointliesonthesphere(16.4),then
Thisisaquadraticequationinrgivingthetwovaluesforrcorrespondingtotwopointscommontothesphereandtheline.Ifthelineisatangentthenthetwovaluesofrmustbeequalandhencethediscriminantmustbezero.
Thelocus(α,β,γ)is
whichisacylinder.Thiscylinderiscalledtheenvelopingcylinderofthesphere.
Envelopingcylinderasalimitingformofanenvelopingcone
Let betheaxisoftheenvelopingcylinder.Anypointonthislineis
(lr,mr,nr).Letthispointbethevertexoftheenvelopingcone.Thentheequationofthe
envelopingconeisT2=SS1.
whichistheequationtotheenvelopingcylinder.
16.4RIGHTCIRCULARCYLINDER
Arightcircularcylinderisasurfacegeneratedbyastraightlinewhichremainsparalleltoafixedstraightlineataconstantdistancefromit.Thefixedstraightlineiscalledtheaxisofthecylinderandtheconstantdistanceiscalledtheradiusofthecylinder.Theequationofarightcircularcylinderwhoseaxisisthestraightline
andwhoseradiusisa.
LetP(x,y,z)beanypointonthecylinder.LetAA′betheaxisofthecylinder.DrawPLperpendiculartotheaxisandPL=a.LetQ(α,β,γ)beapointontheaxisofthecylinder.
Thisistherequiredequationoftherightcylinder.
ILLUSTRATIVEEXAMPLES
Example16.1
Findtheequationofthecylinderwhosegeneratorsareparalleltotheline
andwhoseguidingcurveisx2+y2=9,z=1.
Solution
LetP(x,y,z)beapointonthecylinder.
TheequationsofthegeneratorthroughPandparalleltotheline are
Theguidingcurveis
WhenthegeneratorthroughPmeetstheguidingcurve,
Sincethispointliesonthecurve(16.9),
Thisistheequationoftherequiredcylinder.
Example16.2
Findtheequationofthecylinderwhichintersectsthecurveax2+by2+cz2=1,lx+my+nz=pandwhosegeneratorsareparalleltoz-axis.
Solution
Theequationoftheguidingcurveis
Sincethegeneratorsareparalleltoz-axistheequationofthecylinderisoftheformf(x,y)=0.Theequationofthecylinderisobtainedbyeliminatingzinequation(16.10)
Substitutingthisinax2+by2+cz2=1,weget,
Thisistheequationoftherequiredcylinder.
Example16.3
Findtheequationofthecylinderwhosegeneratorsareparalleltotheline
andwhoseguidingcurveistheellipsex2+2y2=1,z=3.
Solution
Theequationtotheguidingcurveis
Let(x1,y1,z1)beapointonthecylinder.Thentheequationsofthegenerator
throughP(x1,y1,z1)are
Whenthislinemeetstheplanez=3,wehave,
Thispointliesonthecurvex2+2y2=1.
Thelocusof(x1,y1,z1)is
Example16.4
Findtheequationofthesurfacegeneratedbythestraightliney=mx,z=nxand
intersectingtheellipse
Solution
Thegivenliney=mx,z=nxcanbeexpressedinsymmetricalformas
LetP(x1,y1,z1)beanypointonthecylinder.ThentheequationsofthegeneratorthroughPare
Thismeetsthecurve
Whenz=0,
Thispointliesonthecurve
Thelocusof(x1,y1,z1)is
whichistheequationoftherequiredcylinder.
Example16.5
Findtheequationofthecircularcylinderwhosegeneratinglineshavethedirectioncosinesl,m,nandwhichpassesthroughthecircumferenceofthefixedcirclex2+y2=a2onthexozplane.
Solution
LetP(x1,y1,z1)beanypointonthecylinder.Thentheequationsofthegenerator
throughPare
Thismeetstheplaney=0.
Thispointliesonthecurve
Thelocusof(x1,y1,z1)is
whichistherequiredequationofthecylinder.
Example16.6
Findtheequationsoftherightcircularcylinderofradius3withequationsofaxis
as
Solution
Theequationsoftheaxisare
(1,3,5)isapointontheaxis.2,2,–1arethedirectionratiosoftheaxis.
∴directioncosinesare
LetP(x1,y1,z1)beanypointonthecylinder.
Also,PQ2=QL2+LP2
Thelocusof(x1,y1,z1)is
Thisistheequationoftherequiredcylinder.
Example16.7
Findtheequationoftherightcircularcylinderwhoseaxisisx=2y=–zandradius4.
Solution
Theequationsoftheaxisofthecylinderare
LetP(x1,y1,z1)beanypointonthecylinder.TheequationsofthegeneratorthroughPare
Thedirectioncosinesoftheaxisare .
Also,PQ2=QL2+LP2
Thelocusof(x1,y1,z1)is
Example16.8
Findtheequationofthecylinderwhosegeneratorshavedirectioncosinesl,m,nandwhichpassesthroughthecirclex2+z2=a2,y=0.
Solution
LetP(x1,y1,z1)beanypointonthecylinder.TheequationofthegeneratorsthroughPare
Thislinemeetsthecurvey=0,x2+z2=a2
Thispointlieson
Thelocusof(x1,y1,z1)is
Thisistheequationoftherequiredcylinder.
Example16.9
Findtheequationoftherightcircularcylinderwhoseaxisis and
passesthroughthepoint(0,0,3).
Solution
Theequationsoftheaxisofthecylinderare
LetP(x1,y1,z1)beanypointonthecylinder,then
Thelocusof(x1,y1,z1)is
Thisistheequationoftherequiredcylinder.
Example16.10
Findtheequationtotherightcircularcylinderwhichpassesthroughcirclex2+y2+z2=9,x–y+z=3.
Solution
Fortherightcircularcylinder,theguidingcurveisthecirclex2+y2+z2=9,x–y+z=3.Therefore,thedirectionratiosoftheaxisofthecylinderare1,–1,1.LetP(x1,y1,z1)beanypointonthecylinder.ThentheequationsofthegeneratorthroughPare
Anypointonthislineis(r–x1,–r+y1,r+z1).Ifthispointliesonthecircle,then
Eliminatingrfrom(16.16)and(16.17)weget
Simplifying,thelocusof(x1,y1,z1)is
whichistheequationoftherequiredcylinder.
Example16.11
Findtheequationtotherightcircularcylinderofradiusawhoseaxispassesthroughtheoriginandmakesequalangleswiththecoordinateaxes.
Solution
Letl,m,nbethedirectioncosinesoftheaxisofthecylinder.
Theaxispassesthroughtheorigin.LetP(x1,y1,z1)beanypointonthecylinder.
Thelocusof(x1,y1,z1)is
Thisistheequationoftherequiredcylinder.
Example16.12
FindtheequationtotherightcircularcylinderdescribedonthecirclethroughthepointsA(1,0,0),B(0,1,0)andC(0,0,1)astheguidingcurvex2+y2+z2–yz–zx–xy=1.
Solution
TheequationofthesphereOABCis
TheequationoftheplaneABCis
Therefore,theequationofthecircleABCis
Thecentreofthesphereis .
TheequationsofthelineCNare
whichistheaxisofthecylinder.Thedirectionratiosoftheaxisare1,1,1.
Thedirectioncosinesoftheaxisare
Thelocusof(x1,y1,z1)is
whichistherequiredequationofthecylinder.
Example16.13
Findtheequationoftheenvelopingcylinderofthespherex2+y2+z2–2x+4y=1havingitsgeneratorsparalleltothelinex=y=z.
Solution
LetP(x1,y1,z1)beanypointonatangentwhichisparalleltotheline
Therefore,theequationofthetangentlinesare
Anypointonthislineis(x1+r,y1+r,z1+r).Thispointliesinthissphere
Ifequation(16.22)touchesthesphereofequation(16.23),thenthetwovaluesofrofthisequationareequal.
Onsimplifyingwegetthelocusof(x1,y1,z1)as
whichistherequiredequation.
Exercises
1. Findtheequationofthecylinder,whoseguidingcurveisx2+z2–4x–2z+4=0,y=0andwhoseaxiscontainsthepoint(0,3,0).Findalsotheareaofthesectionofthecylinderbyaplaneparalleltothexzplane.
2. Findtheequationofthecylinder,whosegeneratorsareparalleltotheline andpassing
throughthecurvex2+2y2=1,z=0.3. Provethattheequationofthecylinderwithgeneratorsparalleltoz-axisandpassingthroughthe
curveax2+by2=2cz,lx+my+nz=pisn(ax2+by2)+2c(lx+my)–2pc=0.
4. Findtheequationofthecylinder,whosegeneratorsareparalleltotheline andpasses
throughthecurvex2+y2=16,z=0.5. Findtheequationtothecylinderwithgeneratorsparalleltoz-axiswhichpassesthroughthecurve
ofintersectionofthesurfacerepresentedbyx2+y2+2z2=12andlx+y+z=1.
6. Findtheequationofthecylinder,whosegeneratorsintersecttheconicax2+2hxy+by2=1,z=0andareparalleltothelinewithdirectioncosinesl,m,n.
7. Acylindercutstheplanez=0withcurve andhasitsaxisparallelto3x=–6y=2z.
Finditsequation.
8. Astraightlineisalwaysparalleltotheyzplaneandintersectsthecurvesx2+y2=a2,z=0andx2
=az,y=0.Provethatitgeneratesthesurfacex4y2=(x2–az)2(a2–x2).9. Findtheequationofarightcircularcylinderofradius2whoseaxispassesthrough(1,2,3)and
hasdirectioncosinesproportionalto2,1,2.10. Findtheequationoftherightcircularcylinderofradius2whoseaxispassesthrough(1,2,3)and
hasdirectioncosinesproportional2,–3,6.
11. Findtheequationoftherightcircularcylinderofradius1withaxisas
12. Findtheequationoftherightcircularcylinderwhosegeneratorsareparallelto and
whichpassesthroughthecurve3x2+4y2=12,z=0.13. Findtheequationoftherightcircularcylinderofradius4whoseaxisisx=2y=–z.14. Findtheequationoftherightcircularcylinderwhoseguidingcurveisthecirclethroughthepoint
(a,0,0),(0,b,0),(0,0,c).
15. Findtheequationoftheenvelopingcylinderofthespherex2+y2+z2–2x+4y=1havingitsgeneratorsparalleltothelinex=y=z.
16. Findtheenvelopingcylinderofthespherex2+y2+z2–2y–4z=11havingitsgeneratorsparalleltothelinex=–2y=2z.
17. Findtheequationoftherightcylinderwhichenvelopesasphereofcentre(a,b,c)andradiusranditsgeneratorsparalleltothedirectionl,m,n.
Answers
1.10x2+5y2+13z2+12xy+4xz+6yz–36x–30y–18z+36=0
2.3x2+6y2+3z2–2xz+8yz−3=0
4.9x2+9y2+5z2–6xz–12yz–144=0
5.3x2+3y2+4xy–4x–4y–10=0
7.36x2+9y2+17z2+6yz–48xz–9=0
9.5x2+8y2+5z2–4yz–8zx–4xy+22x–16y–14z–10=0
10.45x2+40y2+13z2+36yz–24zx+12xy–42x–280y–126z+294=0
11.10x2+5y2+13z2–12xy–6yz–4zx–8x+30y–74z+59=0
12.27x2+36y2+112z2–36xz–120yz–180=0
13.5x2+8y2+5z2–4xy+4yz+8zx–144=0
14.
15.x2+y2+z2–xy–yz–zx–4x+5y–z–2=0
16.5x2+8y2+8z2+4xy+2yz–4xz+4x–18y–36z=99
17.(l2+m2+n2)[(x–a)2+(y–b)2+(z–c)2–r2]
=[l(x–a)+m(y–b)+n(z–c)]2
Acknowledgements
IexpressmysincerethankstoPearsonEducation,India,especiallytoK.Srinivas,Sojan,Charles,andRameshfortheirconstantencouragementandforsuccessfullybringingoutthisbook.
P.R.Vittal
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