c.p. algebra ii the conic sections index the conics the conics translations completing the square...

C.P. Algebra IIC.P. Algebra II

The Conic Sections The Conic Sections IndexIndex

The ConicsThe Conics

TranslationsTranslations

Completing the SquareCompleting the Square

Classifying ConicsClassifying Conics

The ConicsThe ConicsParabolParabolaa

CircleCircle

EllipseEllipse

HyperbolHyperbolaa

Click on a Click on a PhotoPhoto

Back to IndexBack to Index

The ParabolaThe Parabola

A parabola is A parabola is formed when formed when

a plane a plane intersects a intersects a cone and the cone and the base of that base of that

conecone

ParabolasParabolas

AA Parabola Parabola is a set of points equidistant is a set of points equidistant from a fixed point and a fixed line.from a fixed point and a fixed line.

The fixed point is called the The fixed point is called the focusfocus..

The fixed line is called the The fixed line is called the directrixdirectrix..

Parabolas Around UsParabolas Around Us

ParabolasParabolas

FOCUSFOCUS

DirectrixDirectrix

ParabolaParabola

Standard form of the Standard form of the equation of a parabola equation of a parabola

with vertex (0,0)with vertex (0,0)

•EquatioEquationn

•FocusFocus•DirectriDirectri

xx•AxisAxis

•xx22=4=4pypy

•(0,p)(0,p)•y = -y = -

pp

•yy22=4=4pxpx

•(p,0)(p,0) •y = py = p

To Find pTo Find p4p is equal to the term in front of 4p is equal to the term in front of x or y. Then solve for p.x or y. Then solve for p.

Example:Example:

xx22=24y=24y

4p=244p=24

p=6p=6

Examples for ParabolasExamples for ParabolasFind the Focus and DirectrixFind the Focus and Directrix

Example 1Example 1

y = 4xy = 4x22

xx22= (= (11//44)y)y

4p = 4p = 11//44

p = p = 11//1616

FOCUSFOCUS

(0, (0, 11//1616))

DirectrixDirectrix

Y = - Y = - 11//1616


Example 2Example 2

x = -3yx = -3y22

yy22= (= (-1-1//33)x)x

4p = 4p = -1-1//33

p = p = -1-1//1212

FOCUSFOCUS

((-1-1//1212, 0), 0)

DirectrixDirectrix

x = x = 11//1212


Example 3 Example 3 (try this (try this one on one on your own)your own)

y = -6xy = -6x22

FOCUSFOCUS

????????

DirectrixDirectrix

????????


Example 3Example 3

y = -6xy = -6x22

FOCUSFOCUS

(0, -(0, -11//2424))

DirectrixDirectrix

y = y = 11//2424


Example 4 Example 4 (try this (try this one on one on your own)your own)

x = 8yx = 8y22

FOCUSFOCUS

????????

DirectrixDirectrix

????????


Example 4Example 4

x = 8yx = 8y22

FOCUSFOCUS

(2, 0)(2, 0)

DirectrixDirectrix

x = -2x = -2

Parabola ExamplesParabola Examples

Now write an equation in Now write an equation in standard form for each of standard form for each of

the following four the following four parabolasparabolas

Write in Standard FormWrite in Standard Form

Example 1Example 1Focus at (-4,0)Focus at (-4,0)

Identify equationIdentify equationyy2 2 =4px p = -4=4px p = -4yy2 2 = 4(-4)x= 4(-4)xyy2 2 = -16x= -16x


Example 2Example 2With directrix y = 6With directrix y = 6Identify equationIdentify equationxx2 2 =4py p = -6=4py p = -6xx2 2 = 4(-6)y= 4(-6)yxx2 2 = -24y= -24y


Example 3 (Now try this Example 3 (Now try this oneone

on your own)on your own)

With directrix x = -1With directrix x = -1

yy2 2 = 4x= 4x


Example 4 (On your own)Example 4 (On your own)

Focus at (0,3)Focus at (0,3)

xx2 2 = 12y= 12y

Back to ConicsBack to Conics

CirclesCircles

A Circle is formed when a A Circle is formed when a plane intersects a cone plane intersects a cone

parallel to the base of the parallel to the base of the cone.cone.

CirclesCircles

Standard Equation of a Standard Equation of a Circle with Center (0,0)Circle with Center (0,0)

x 2 y 2 r 2

Circles & Points of Circles & Points of IntersectionIntersection

Distance formula used Distance formula used to find the radiusto find the radius

(x1 x2 )2 (y1 y2)2 r

CirclesCirclesExample 1Example 1

Write the equation of the circle with the Write the equation of the circle with the point (4,5) on the circle and the origin point (4,5) on the circle and the origin as it’s center.as it’s center.

Example 1Example 1

Point (4,5) on the circle and the origin Point (4,5) on the circle and the origin as it’s center.as it’s center.

(x1 x2 )2 (y1 y2)2 r

(4 0)2 (5 0)2 r

16 25 r

41 r

x 2 y 2 41

Example 2Example 2Find the intersection points on the graph of Find the intersection points on the graph of

the following two equationsthe following two equations

x 2 y 2 25

y 2x 2

x 2 (2x 2)2 25

x 2 4x2 8x 4 25

5x2 8x 4 25

5x2 8x 21 0

(5x 7)(x 3) 0

(5x 7) 0

(x 3) 0

x 3

x 7

5

Now what??!!??!!??Now what??!!??!!??



x 2 y 2 25

y 2x 2

x 3

x 7

5

Plug these Plug these in for xin for x.



x 2 y 2 25

y 2x 2

y 2(7

5) 2

y 24

5

7

5,24

5

x 7

5

x 3

y 2( 3) 2

y 4

3, 4 Back to ConicsBack to Conics

EllipsesEllipses

EllipsesEllipses

Examples of EllipsesExamples of Ellipses

EllipsesEllipses

Horizontal Major AxisHorizontal Major Axis

FOCIFOCI(-c,0) & (c,0)(-c,0) & (c,0)

CO-VERTICESCO-VERTICES(0,b)& (0,-b)(0,b)& (0,-b)

CENTER (0,0)CENTER (0,0) VerticesVertices(-a,0) & (a,0)(-a,0) & (a,0)x2

a2y2

b2 1

EllipsesEllipses

Vertical Major AxisVertical Major Axis

FOCIFOCI(0,-c) & (0,c)(0,-c) & (0,c)

CO-VERTICESCO-VERTICES(b, 0)& (-b,0)(b, 0)& (-b,0)

VerticesVertices(0,-a) & (0, a)(0,-a) & (0, a)

CENTER (0,0)CENTER (0,0)

x 2

b2 y 2

a2 1

Ellipse NotesEllipse Notes Length of major axis = a (vertex & Length of major axis = a (vertex &

larger #)larger #)

Length of minor axis = b (co-vertex Length of minor axis = b (co-vertex & smaller#)& smaller#)

To Find the foci (c) use: To Find the foci (c) use:

cc22 = a = a2 -2 - b b22

Ellipse ExamplesEllipse ExamplesFind the Foci and VerticesFind the Foci and Vertices

x2

144y 2

1690

avertices

a13

vertices (0,13),(0, 13)

c 2 a2 b2

c 2 169 144

c 2 25

c 5

foci(0,5),(0, 5)

Ellipse ExamplesEllipse ExamplesFind the Foci and VerticesFind the Foci and Vertices

x 2

81y 2

90

avertices

a9

vertices (9,0),( 9,0)

c 2 a2 b2

c 2 81 9

c 2 72

c 72

foci( 72,0),( 72,0)

Write an equation of an ellipse Write an equation of an ellipse whose vertices are (-5,0) & (5,0) whose vertices are (-5,0) & (5,0) and whose co-vertices are (0,-3) and whose co-vertices are (0,-3)

& (0,3). Then find the foci.& (0,3). Then find the foci.

a5

a2 25

b 3

b2 9

x 2

25y 2

91

c 2 a2 b2

c 2 25 9

c 2 16

c 4

foci(4,0),( 4,0)

Write the equation in standard Write the equation in standard form and then find the foci and form and then find the foci and

vertices.vertices.

49x 2 64 y2 3136

c 2 a2 b2

c 2 64 49

c 2 15

c 15

foci( 15,0),( 15,0)

49x 2

3136

64y2

3136

3136

3136

x 2

64y2

491

vert.(8,0),( 8,0)

Back to the Back to the ConicsConics

The HyperbolaThe Hyperbola

Hyperbola ExamplesHyperbola Examples

Hyperbola NotesHyperbola NotesHorizontal Transverse AxisHorizontal Transverse Axis

Center Center (0,0)(0,0)

Vertices (a,0) Vertices (a,0) && (-a,0)(-a,0)

Foci (c,0) &Foci (c,0) & (-c, 0)(-c, 0)

AsymptotAsymptoteses


EquationEquation

x 2

a2 y2

b2 1

Foci : c2 a2 b2


To find To find asymptotesasymptotes

y b

ax

y bax

Hyperbola NotesHyperbola NotesVertical Transverse AxisVertical Transverse Axis

Center Center (0,0)(0,0)

Vertices (a,0) Vertices (a,0) && (-a,0)(-a,0)

Foci (c,0) &Foci (c,0) & (-c, 0)(-c, 0)

AsymptotAsymptoteses


EquationEquation

y 2

a2 x2

b2 1

Foci : c2 a2 b2


To find To find asymptotesasymptotes

y a

bx

y abx

Write an equation of the Write an equation of the hyperbola with foci (-5,0) & (5,0) hyperbola with foci (-5,0) & (5,0)

and vertices (-3,0) & (3,0)and vertices (-3,0) & (3,0)

c 2 a2 b2

52 32 b2

25 9 b2

b2 16

x 2

9y2

161

a = 3 c = 5a = 3 c = 5

Write an equation of the Write an equation of the hyperbola with foci (0,-6) & (0,6) hyperbola with foci (0,-6) & (0,6)

and vertices (0,-4) & (0,4)and vertices (0,-4) & (0,4)

c 2 a2 b2

62 42 b2

36 16 b2

b2 20

y 2

16x2

201

a = 4 c = 6a = 4 c = 6

The ConicsThe Conics

TranslationsTranslations

BackBack

What happens when the What happens when the conic is NOT centered on conic is NOT centered on (0,0)?(0,0)?

NextNext

TranslationsTranslationsCircleCircle

NextNext

(x h)2 (y k)2 r 2

TranslationsTranslationsParabolaParabola

NextNext

oror

Horizontal AxisHorizontal Axis

Vertical AxisVertical Axis

(y k)2 4 p(x h)

(x h)2 4 p(y k)

TranslationsTranslationsEllipseEllipse

NextNext

oror

(x h)2

a2 (y k)2

b2 1

(x h)2

b2 (y k)2

a2 1

TranslationsTranslationsHyperbolaHyperbola

NextNext

oror

(x h)2

a2 (y k)2

b2 1

(y k)2

a2 (x h)2

b2 1

TranslationsTranslationsIdentify the conic and graphIdentify the conic and graph

NextNext

(x 1)2 (y 2)2 32

r=r= 33 centercenter (1,-2)(1,-2)


NextNext

(x 2)2

32 (y 1)2

22 1


NextNext

(x 3)2

12 (y 2)2

32 1

centecenterrasymptotasymptoteses

verticesvertices


(x 2)2 4( 1)(y 3)

centecenterr

ConiConicc



Here are the steps for completing the Here are the steps for completing the squaresquare

StepsSteps

1)1) Group xGroup x22 + x, y + x, y22+y move constant+y move constant

2)2) Take # in front of x, ÷2, square, Take # in front of x, ÷2, square, add to both sidesadd to both sides

3)3) Repeat Step 2 for y if neededRepeat Step 2 for y if needed

4)4) Rewrite as perfect square binomialRewrite as perfect square binomialNextNext


Circle: xCircle: x22+y+y22+10x-6y+18=0+10x-6y+18=0

xx22+10x+____ + y+10x+____ + y22-6y=-18-6y=-18

(x(x22+10x+25) + (y+10x+25) + (y22-6y+9)=--6y+9)=-18+25+918+25+9

(x+5)(x+5)22 + (y-3) + (y-3)22=16=16

Center (-5,3)Center (-5,3) Radius = 4Radius = 4 NextNext


Ellipse: xEllipse: x22+4y+4y22+6x-8y+9=0+6x-8y+9=0

xx22+6x+____ + 4y+6x+____ + 4y22-8y+____=-9-8y+____=-9

(x(x22+6x+9) + 4(y+6x+9) + 4(y22-2y+1)=--2y+1)=-9+9+49+9+4

(x+3)(x+3)22 + (y-1) + (y-1)22=4=4

(x 3)2

4

(y 1)2

11

C: (-3,1)C: (-3,1)

a=2, b=1a=2, b=1

IndexIndex


Given in General Given in General FormForm

Ax 2 Bxy Cy 2 Dx Ey F 0

NextNext

B2 4AC 0

B2 4AC 0

B2 4AC 0


Given in General Given in General FormForm

Ax 2 Bxy Cy 2 Dx Ey F 0

ExamplesExamples

B2 4AC 0

B2 4AC 0

B2 4AC 0


Given in general Given in general form, classify the form, classify the

conicconic

5x2 2y2 20x 4y 24 0

A5

B0

C 2

B2 4AC

02 4(5)(2)

40

EllipsEllipseeNextNext



conicconic

y 2 8x 12y 0

A0

B0

C 1

B2 4AC

02 4(0)(1)

0

ParabolParabolaa

NextNext



conicconic

24x 2 18y2 18 0

A 24

B18

C 0

B2 4AC

02 4( 24)(18)

1728

HyperboHyperbolala

NextNext



conicconic

16 xy0

A0

B 1

C 0

B2 4AC

( 1)2 4(0)(0)

1

HyperboHyperbolala


Classifying ConicsClassifying ConicsGiven in General Given in General FormForm

ThenThen

ORORIf A = CIf A = C

EllipsEllipsee

CirclCirclee

BackBack


ThenThen

BackBack


ThenThen

HyperbolaHyperbola

BackBack

c.p. algebra ii the conic sections index the conics the conics translations completing the square...

Documents

standard formexample

directrix y

intersection points

fixed point

ownwith directrix x

circlesstandard equation

fixed line

conicscirclesa circle