123328034 6 coordinate geometry doc

7/22/2019 123328034 6 Coordinate Geometry Doc

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6. COORDINATE GEOMETRY

Unit 6.1 : To Find the distance between two oints [BACK TO BASICS]

A( 11 ,yx ) and B( 22 ,yx ) : AB =2 2

2 1 2 1( ) ( )x x y y + .

Eg. 1 Given two points A(2,3) and B(4,7)

Distance of AB = 2 2(4 2) (7 3) +

= 4 1+

= 2! "nit.

E1. #(4,$) and %(3,2)

#% =

& 1! '

E2. ($,!) and ($,2)

&2'

E3. *(7,1) and +(2,$)

& 41 '

E4. (1!,) and -(4,2)

& $2 '

E$. (/4,/1) and 0(/2,1)

& 1 '

oe ca55enging %"estions6.E1. *e distance etween two points A(1, 3) and

B(4, 8) is $. 9ind te possi5e va5es of 8.

7, /1

E2. *e distance etween two points #(/1, 3) and

%(8, ) is 1!. 9ind te possi5e va5"es of 8.

7, /

E3. *e distance etween two points (/2, $) and

(1, 8) is 1! . 9ind te possi5e va5es of 8.

, 4

E4. *e distance etween two points ;(/1, p) and

(!, !) and #(8, 28) is te

sa?e as te distance etween te points A(/4, 3) and

B(1, /7), find te possi5e va5"es of 8.

k = 5, -5

7


2/17

Unit 6.2 : Division of a Line Segment

6.!.1 To "ind the #id$oint o" Two Gi%en &oints.

9o?"5a : idpoint =

++

2,

2

2121 yyxx

Eg. #(3, 2) and %($, 7)

idpoint, =

++

2

72,

2

$3

= (4 ,2

:)

E1 #(/4, ) and %(, !)

(2, 3)

E2 #(, 3) and %(2, /1)

(4, 1)

E3 #(!,/1), and %(/1, /$)

(/ @ , /3)

6.!.! Di%ision o" a 'ine (e)#ent

% divides te 5ine seg?ent # in te atio PQ :QR= ? : n. #(x,y), (x,y)

% (x,y)= ++++ nmmyny

nm

mxnx2121 ,

(NOT : St!"ents a#e st#ong$% a"vise" to s&et' a $ine segment )efo#e a**$%ing te fo#m!$a+

Eg1. *e point # intena55 divides te 5ine seg?ent

oining te point (3,7) and C(,2) in te atio 2 : 1.

9ind te coodinates of point #.

# =

++

++

12

)2(2)7(1,

12

)(2)3(1

=

3

11,

3

1$=

11$,

3

E1. *e point # intena55 divides te 5ine seg?ent

oining te point (4,$) and C(/,/$) in te atio

1 : 3. 9ind te coodinates of point #.

$1,

2

More Exercise : The Ratio Theorem

n?

#(1,

1) (

2,

2),(- %+

n

?

(2,

2)

#(1,

1)

,(- %+

1

2

C(, 2)

(3, 7)

/(- %+


3/17

(NOT : St!"ents a#e st#ong$% a"vise" to s&et' a $ine segment )efo#e a**$%ing te fo#m!$a+

E1. divides #% in te atio 2 : 1. 9ind te

coodinates of if

(a) #(1, 2) and %( /$, 11)

() #(/4, 7) and %(, /$)

(a) (/3, ) () (4 , /1)

E2. # divides AB in te atio 3 : 2. 9ind te

coodinates of # if

(c) A(2, /3) and B( /, 7)

(d) A(/7, $) and B(, /$)

(a) (/4, /3) () (2 , /1)

E3. is a point tat 5ies on te staigt 5ine s"c

tat 3 = . f te coodinates of te points

and ae (4,$) and (/,/$) espective5, find te

coodinates of point .

3 =

MS

RM=3

1, : = 1 : 3

Ans :

2

$,1

E4. # is a point tat 5ies on te staigt 5ine *+ s"c

tat 3*# = 2#+. f te coodinates of te points * and

+ ae (/,7) and (1,/3) espective5, find te

coodinates of point #.

(/$, 3)

E$. *e points #(3, p), B(/1, 2) and F(,7) 5ie on a

staigt 5ine. f # divides BF intena55 in te atio? : n , find (a) ? : n , () te va5"e of p.

(a) 2 : 3 () p = 4

E. (x, y) , divides te points #(28, 8) and

%(2x, 4y) in te atio 3 : $. Epessxin te?s of y.

(Ans : = 4)

Unit 6.* To Find A+eas o" &o,-)ons


4/17

Aea of a po5gon =2

1

1 2 3 1

1 2 3 1

...

...

x x x x

y y y y

Note : Te a#ea fo!n" 0i$$ )e *ositive if te 'oo#"inates of te *oints a#e 0#itten in te anti1'$o'&0ise

o#"e# an" negative if te% a#e 0#itten in te '$o'&10ise o#"e#.

Ea?p5e 1 : Fa5c"5ate te aea of a tiang5e given :

E1. #(!, 1), %(1, 3) and (2,7)

Aea of H #% =2

1 ! 1 2 !

1 3 7 1

=

= 1 "nit 2

1. #(2,3), %($,) and (/4,4)

Aea of H #% =

17

2"nit2

2. *e coodinates of te tiang5e ABF ae ($, 1!), (2,1)and (, 8) espective5. 9ind te possi5e va5"es of 8,

given tat te aea of tiang5e ABF is 24 "nits2.

8 = 3 , 3$

3. *e coodinates of te tiang5e * ae (4, 3), (/1, 1) and(t, /3) espective5. 9ind te possi5e va5"es of t , given tat

te aea of tiang5e * is 11 "nits2.

t = ! , /22

ii) Aea of a I"adi5atea5 =2

1

14321

14321

yyyyy

xxxxx

1. #(1,$), %(4,7), (,) and (3,1).

Aea of #% =

= "nit 2

2. #(2, /1), %(3,3), (/1, $) and (/4, /1).

&27'

Note : If te a#ea is e#o ten te *oints a#e 'o$$inea#.

1. Given tat te points #($, 7), %(4, 3) and (/$, 8) ae

co55inea, find te va5"e of 8.

2. ow tat te points ;(4, ),


5/17

8= 33

Unit 6. : E/0ations o" (t+ai)ht 'ines

*e EI"ation of a taigt 5ine ?a e epessed in te fo55owing fo?s:

i) *e genea5 fo? : a J J c = !

ii) *e gadient fo? : y = mx c ! m = "ra#ient , c = y-interce$t

iii) *e intecept fo? :a

xJ%

y= 1 , a = x-interce$t , % = y-interce$t

a+ If given te g#a"ient an" one *oint:

1yy = )( 1xxm

Eg. 9ind te eI"ation of a staigt 5ine tat passes

to"g te point (2,/3) and as a gadient of4

1.

1yy = )( 1xxm

)2(4

1)3( = xy

144 =xy

E1. 9ind te eI"ation of a staigt 5ine tat passes

to"g te point ($,2) and as a gadient of /2.

= /2 J 12


to"g te point (/,3) and as a gadient of4

3.

4 = 3 J 3

b I" two oints a+e )i%en :

Cote : 0o" ?a find te gadient fist, ten "se

eite (a) y = mx c

> () y & y'= m( x & x')

>

(c)1

1

xx

yy

=12

12

xx

yy

Eg. 9ind te eI"ation of a staigt 5ine tat passes

to"g te points (/3, /4) and (/$,)

)3(

)4(

xy = )3($ )4(

1:$ = xyE1. 9ind te eI"ation of a staigt 5ine tat passes

to"g te points (2, /1) and (3,!)


to"g te points (/4,3) and (2,/$)

1

Gadient = m

/(-3 %3+


6/17

= / 3 4 J 3 J7 = !

c+ Te -1inte#'e*t an" te %1inte#'e*t a#e given:

m= /

errce$tx

erce$ty

int

int

EI"ation of taigt


7/17

x 2y & = 0 1x =

Unit 6.7 /a#a$$e$ Lines an" /e#*en"i'!$a# $ines

6.2.1 &a+a,,e, ,ines3 1m 4 2m6.2.! &e+endic0,a+ ,ines3 21mm 4 $1

Unit 6.7.3 Dete#mine 0ete# ea' of te fo$$o0ing *ai#s of $ines a#e *a#a$$e$.

Eg. = 3 2 and 3 = 4

= 3 2 , ?1 = 3

3 = 4

= 3 4 , ?2 = 3

ince ?1= ?2 , te two 5ine ae paa55e5 .

1. = 2 J$ and 4 J 2 = $

C

2. 3 3 = 7 and J = $

C

3. 2 3 = $ and = 4 J

0

4. 3 = 12 and = 3 J 2

0

$. 43 2

x y = and = / 3

C

Unit 6.7.2 Dete#mine 0ete# ea' of te fo$$o0ing *ai#s of $ines a#e *e#*en"i'!$a#.

Eg. = 3 2 and J 3 = 4

= 3 2 , ?1 = 3

J 3 = 43 = J 4

1 4

3 3y x= + , ?2 =

1

3

ince ?1. ?2 = 13 13 =

,

*e two given 5ines ae pependic"5a .

1. = 2 J$ and 4 J 2 =

C

2. 3 = 2 2 and 2 J 3 = 1

C

3. 3 = 2 and J 2 = $

0

4. = 2 / 3 and 43

x y

=

$. 13 4

x y

= and J 3 = !

3


8/17

0 0

6.7.2 A**$i'ations (m3.m25 8 3+

E5.1 (&M !77. Diaga? 1 sows a staigt

5ine #% wit te eI"ation 12 4

x y+ = . 9ind te

eI"ation of te staigt 5ine pependic"5a to #%

and passing to"g te point %.

Answe:

y = 3 x 4

E.2. Diaga? 2 sows a staigt 5ine #% wit te

eI"ation 1 2

x y+ = . 9ind te eI"ation of te

staigt 5ine pependic"5a to #% and passing

to"g te point #.

Answe:

y = x & '/

E.3 Diaga? 3 sows a staigt 5ine wit te

eI"ation J 2 = . 9ind te eI"ation of te

staigt 5ine pependic"5a to and passing

to"g te point .

Answe:

E.4. Diaga? 4 sows a staigt 5ine AB wit te

eI"ation 2 3 = . 9ind te eI"ation of te

staigt 5ine pependic"5a to AB and passing

to"g te point B.

Answe:

4

x

y

.

Q

P

Diaga? 1

x

y

.

Q

P

Diaga? 2

x

y

.

R

S

Diaga? 3x

y

.

Diaga? 4


9/17

y = 2x & '2 2y = x & 6

6.7.2 A**$i'ations (m3.m25 8 3+ 8 mo#e e-e#'ises

E.$ Diaga? $ sows a staigt 5ine #% wit teeI"ation 4 J 3 = 12. 9ind te eI"ation of te

staigt 5ine pependic"5a to and passing

to"g te ?idpoint of .

Answe:

4xy = /

E.. Diaga? sows a staigt 5ine AB wit teeI"ation 1

4

x y = . 9ind te eI"ation of te

pependic"5a isecto of te 5ine AB.

Answe:

2x y =

E.7. 9ind te eI"ation of te staigt 5ine tatpasses to"g te point ( 1, 2) and is pependic"5a

to te staigt 5ine J 3 J = !.

y = x & '

E. 9ind te eI"ation of te staigt 5ine tatpasses to"g te point (3, !) and is pependic"5a

to te staigt 5ine 3 2 = 12.

2xy =

E. 9ind te eI"ation of te staigt 5ine tat

passes to"g te oigin > and is pependic"5a to

te staigt 5ine tat passes to"g te points

#(1, 1 ) and %(/3,7).

y = 3 x

E. 1! 9ind te eI"ation of te staigt 5ine tat

passes to"g te point (/2,4) and is pependic"5a

to te staigt 5ine wic passes to"g te oigin

> and te point (, 2).

y = -x

$

x

y

.

R

S

Diaga? $ x

y

.

Diaga?


10/17

Unit 6.6 9!ation of a Lo'!s

Note : St!"ents UST )e a)$e to fin" "istan'e )et0een t0o *oints [ !sing /%tago#as Teo#em]

*A; : *o 9ind te eI"ation of te 5oc"s of te ?oving point # s"c tat its distances of # fo? te points

% and ae eI"a5.

Eg 1. %(, /$) and (1,)


11/17

y =2x - ' x y = 4

TA(8 : To "ind the e/0ation o" the ,oc0s o" the #o%in) oint & s0ch that its distances "+o#

the oints A and 9 a+e in the +atio # : n(Note : S&et' a "iag#am to e$* %o! !sing te "istan'e fo#m!$a 'o##e't$%+

Eg 1. A(/2,3), B(4,) and ? : n = 1: 2


12/17

x2y2 '0x & '4y 2 = 0 x2y2 'x 2y 5 = 0

S/


13/17

1

2$

k=

1

2 1$

k=

7. (2!!) Diaga? $ sows te staigt 5ine AB

wic is pependic"5a to te staigt 5ine FB at te

point B.

*e eI"ation of FB is = 2 1 .

9ind te coodinates of B.

(2, )

.Diaga? sows te staigt 5ine #% wic is

pependic"5a to te staigt 5ine % at te point %.

*e eI"ation of % is = 4 .

9ind te coodinates of %

%($, 1)

.(2!!4) *e point A is (/1, 2) and B is (4, ). *e

point # ?oves s"c tat #A : #B = 2 : 3. 9ind te

eI"ation of 5oc"s of #. &3 ?a8s'

15x25y250x'2y & '=0

1!. *e point is (3, /$) and is (!, 1). *e point

# ?oves s"c tat # : # = 2 : 1. 9ind te

eI"ation of 5oc"s of #. &3 ?a8s'

1x2y22x & y & '0 = 0

11.*e point A is (, /2) and B is (4, ). 9ind te

eI"ation of te pependic"5a isecto of AB.

&3 ?a8s'

12. *e point is (2, /3) and is (4, $). *e point

# ?oves s"c tat it is a5was te sa?e distance

fo? and fo? . 9ind te eI"ation of 5oc"s of

#. &3 ?a8s'

x

y

O

A( >+

C

Diaga? $B

?

?

?

x

y

O

/( 6+

=

Diaga? ,

?

?

?


14/17

2y = x & 2

x4y = ?

SPM Questions (Coordinate Geometry) Paper 2

Cote : So$!tions to tis 9!estion )% s'a$e "#a0ing 0i$$ not )e a''e*te".

3. (S/ 26 /2 ,@+

Diaga? 3 sows te tiang5e.wee .is te oigin. #oint @5ies on te staigt 5ine.

(a) Fa5c"5ate te aea, in "nits2, of tiang5e.. &2 marks'

() Given tat@ : @ = 3 : 2, find te coodinates of @. &2 marks'

(c) A pointP?oves s"c tat its distance fo? pointis a5was twice its distance fo? point.

(i) 9ind te eI"ation of 5oc"s ofP,

(ii) Lence, dete?ine wete o not tis 5oc"s intecepts tey/ais. & marks'

2. (S/ 27 /2 ,@+n Diaga? $, @= !! and te eI"ation of te staigt 5ine@is 2yJxJ = !.

(a) 9ind

(i) te eI"ation of te staigt 5ine

(ii) te coodinates of &$ marks'

() *e staigt 5ineis etended to a pointAs"c tat : A = 2 : 3.

9ind te coodinates ofA. &2 marks'

(c) A pointP?oves s"c tat its distance fo? pointis a5was $ "nits.9ind te eI"ation of te 5oc"s ofP. &3 marks'

1!!

x

y

O

A(1 >+

Diaga? 3

C

?

?

?

B(6 12+

x

y

O

(-4, 6)

Diaga? $B

?

?

?

C


15/17

. (S/ 2> /2 ,2+

Diaga? 1 sows a staigt 5ine @Awic ?eets a staigt 5ine at te point D.

*e point @5ies on tey/ ais.

(a) -ite down te eI"ations ofin te fo? of intecepts. & 1 mark '

() Given tat 2A = A, find te coodinates ofA> & 2 marks'

(c) Given tat @Ais pependic"5a to,find tey/intecept of @A. & 3 marks'

>. (S/ 2 /2 ,33+

A pointP?oves a5ong te ac of a cic5e wit cente(2, 3). *e ac passes to"g Q(/2, !) and

($, 8).

(a) 9ind

(i) te eI"ation of te 5oc"s ofP,

(ii) te va5"es of k> & marks'

() *e tangent to te cic5e at point Q intesects tey/ is at point T.

9ind te aea of tiang5e .QT> &4 marks'

7. (S/ 22 /2 ,33+

Given tat( 1 , 2 ) and(2, 1) ae two fied points. #ointP?oves s"c tat te atio ofPtoPis

a5was 1 : 2.

(a) ow tat te 5oc"s ofPis 2J 2J4 J J$ = !. &2 marks'

() ow tat te point @(!, /$) 5ies on tis 5oc"s. &2 marks'

(c) 9ind te eI"ation of te 5ine@> &3 marks'

(d) Given tat te 5ine@c"ts te 5oc"s ofPagain at pointA> 9ind te coodinates ofA> &3 marks'

1!1

x

y

O

A( 6+

Diaga? 1

C

?

?

?

B(@ +


16/17

SPM Questions (Coordinate Geometry) Paper 1

1. !77**e eI"ations of two staigt 5ines ae 1

$ 3

y x+ = and $ = 3 J 24. Dete?ine wete te

5ines ae pependic"5a to eac ote. &'

! !77 Diaga? 4 sows a staigt 5ine #% wit te eI"ation 12 3

x y+ = . 9ind te eI"ation of te

staigt 5ine pependic"5a to #% and passing to"g te point %. &3 marks'

12

33

y x= +

*.!77 *e point A is (/1, 2) and B is (4, ). *e point # ?oves s"c tat #A : #B = 2 : 3. 9ind te

eI"ation of 5oc"s of #. & marks'15x25y250x'2y'=0

Epess$in te?s of k.

1!2

. !772*e fo55owing info?ation efes to te eI"ations of two staigt 5ines, K; and *, wic ae

pependic"5a to eac ote.

1

2$

k=

x

y

O

,

/

Diaga? 4


17/17

1!3

2. !776Diaga? $ sows te staigt 5ineABwic is pependic"5a to te staigt 5ine CBat te point.

*e eI"ation of CBis % 5 2- 8 3 .

9ind te coodinates ofB. (2, )

x

y

O

A( >+

C

Diaga? $B

?

?

?

123328034 6 coordinate geometry doc

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