JEE Questions Circle

8/15/2019 JEE Questions Circle

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<

R

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R

1

+

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-

R

R

3

s

(>

-

-

a

)

s

-

b

)

(

.t -

c

)

=

R

1

+

R

,

+

R

,

)

(R

,

)

(R

,

)

(R

1

)

.

j

R

; R

,

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(R

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.

I ~ = R1 11, R

1

(11

1

~ R +R

3

)

(11,+11,+11,)'

The [';iven cHck

;,

2x(.r-a)

4

y(2y- l r l=O

or

xlr-a)+ y(J•-h. 2)=0

'"

x

2

-

1

J

" ' byl2=0

...(I)

Lot

oftlw cborJ

thmu 'h

(a. h 12)

he

Oi,ccld

at ( lr. n).

I

hen the equation

of

the

chord

having {ir.

01 as

mid poim

T=S

1

+O· v - - ( x

. '

" '

/ 1

'>"+0)- it·

-0-

<1h -0

.

·I

.

I

' o \ hy a ,

11-- •x h=h

-air

' 2; 4 2

...(2)

'Jo\\. (2) \\

l ra."

Li>rough ("·

h

12) if

.,

> ) b b

a

..

'-'

>

1 - a

~ a · .

~ - a 1

.)

2 2

'

'

3

Q b-

h - . - - a / J + - + - · =II

"

...

3)

Accordtng to the eiven c o n d i l i ~ n . 11)

rHU't

IM\C two

dJ>ltnd real

roms

I his i< p o s > J ~ l e if lhe rli<o l iminanl nl"

(31

is greater than 0.

That;,,

i l

" , ~ · · a b ]

" - · l2+H Al

o· _ h' >

- _ ,

4

l

Tile equation ol.thc

Lirde an

the li:>c

joining

the pomt<

3(3

7) and /J 6.

5) '

d ;>meter

; ,

(x-3) (x 6 1 (I ' 7)(l - -5)-11 ...(I)

and the c ~ u a - i o n ni" Ihe lin.• e ' " ' " ~ t."e pointsA 13. 7) "nd

R (0. 5)

is

" 7-5

J -1= - < ' l

1 - 0

2x 13y-27=0

...

(2)

Now tho cqnotwr. oC t;,,.,, J' , , t . c i r ~ k ' possing th·c'.Lg.lJ the

r<>in ol"tntcrsecticn of ( I ) and {2) is

S , ) ,P-0

=>

l.t 3)(>. -6) - (y-7) ( •

-5)

·1 . (2x+Jy-C7)-0

· - 6x -3x 113+y ' -5y-

7)

1

>.<+

) _ , + 3 1 - . " - n J . ~ o

0

s,=x·T,

l- '(2 ' ,-9)+y(:li .-12J

f (5

-27 .)=0 ...

(J)

Again tlte circle. which

Cli S

the member.;

of

Glmil)

drcks

is

s, =x'- y '- ,J.< - 6 y - J n ..

•nd the eq1mtion ofcommon chord to Circles S

1

and S,

s

-s,-o

=>

{x(2/,

?) ' l ' (J) . 12)+(53-271.)}

- ( - 4 x - 6 y - 3

xC i . -9 1 ) r ; J

+(13-27), .. )

2./o,x -5x

+

31-.l

-6y+56

-27),

=I)

=> ( - 5 x - 6 y + 5 6 1 + ) . ( 2 , + 3 y - 2 7 ) ~ 0

w 1ich

rcpre<ents equations ofrvm straight lines pa.,

through the fixed point whose coordinates are obtai

by solving the tvco eqllations

5r 1Gy-56=0

ond2x + 3y-27 =0

Soh-ing for

x

andy, we get

x

= 2 and

y

= 23 I 3.

Tv.o cJrcies touch

cac 1

otht'rcxtcrn"lly lf(

1

c,

=

•;

and i n ~ r n a l l '

i f r

, I '

0

- "· - r

0

and ~ / - • h - 2 y = - 1

(giv

=>

c

1

(2.1) an•

r

1

=I

, ,

and

x - + y - - 1 2 . ~ - 8 1 = - 3 6

(giv

C,(fi·1) and r , = ~

The di>tancc bcm-ccn t h ~ centre< is

2) ' - (4

ThereiC>re. l ~ e c i t t b touch each other

ntorrolly

ond

the p0mt

nt"

" ' " " " ' ~ : •he > "' :' ':• ·he line joi:ling

two

c e n t r ~ s

i n t e n n l l ~ in the ratio ofthctr radii. I ; 4.

lx<i•-1-x2

TI1ercfore.

x

1

=

I -

4 5

h4-4 , ·1

R

I + ~

Agmn

lD

detellllin• the equation<

of common

tange

touching the eire "' m Ji>Linc\ poinlo. we lnow that

tangems pass

thr<H•gh

a

po11n

"·l1ich di,·ides

'he

joining tlle

\Wo

centre' cxtemally

in

the ratio of<heir m

i

e. I :

4.

lx6·--h2 -2 2

x, =

=

. I

-4

-3 3

ence.

and J , ~ l x 4 - 4 x i _

, ' '

\ow,

lel"'

be the slope

~ f t t e

t : u t ~ ; e m

Th''

l i n ~

r ~ " '

through (2

.'

3. io

~ . -G=m{x-213)

,

y m x = m ~ O

3

but for

any

an gent. p =r is >a tstied.

Thcrcfore. -

2

m+(

2

/

31

" ' 1 rc,

=(2.l)andr,

~ l l

}1 +

m



C

4U

1

-:

m

,(

2

- ·

) :

;

0

1

- ,

m

=

v

l

~

m ·

'

O

m

~

7

lle

nc

e

,

th

e e

qu

at

io

n,

o

fth

e

tv .

o

ta

ng

en

t

'

y

~

O

a

nd

v=

2

4

[x

-

3

:)

0

y

=

O

an

d

7y

-

-2

4

x+

l

6=

0

l

S

.

'

(

J

,,

S

: ,

x

)

-

l2

-

- /

'

-

-

;

-

y

=

O

St

nc

e.

y

+

x

-0

b

bc

ct

_,

m

o

c

ho

rd

s

of

t

hf

s

c

irc

lo

,

mi

d-

po

in

t<

o

[ t

he

c

ho

r<

ls m

u

s

t t>

c

o

f t

he

fo

rm

a.

-

).

'

(«

, --

-u)

i:

'<J

ua

lio

n

o

f th

e

ch

o

rd

a

. -

a

)

as

m

i

d-p

o

in

ts

is

T

=

S

1

~

=

>

)

(

-a

;

)-

C

+

f2

u

_

}x

+

U

)

f

-=

-

f2

a

}

r

:J

'r

i+

-

-

J

2

a

l

r

l

-.

fi

a

'i

=

<1

+

(

-a

)

-

r

-

i

t

"

)

4x

<:t

-

4

_ru

-

(l

-

r

x

-

(l

+

lia

«

- (

1

-

-v

2a

) y

+

(

1-

--

./2

a)

4c

:t

' +

4

u

'

- (1

+ J

a1

. 2

o.

+

(

1

- -

-Jl

a)

· 2

a

'-'>

4a

x -

4o

.y

-

(I

+

. f

i

a)

x

-

(1

- .f

ia

)

y

=

8a

2

-

(l

+-

'

"2

a

)a

+

(l

-.

Ji

a

)a

B

u

t t

h i

s c

ho

rd

w

ill

p

a

ss

th

ro

ug

h

the

>

"

'in

[

l

+

-h

a

1

-

.fi

.a

'j

0

.

.

'

4

u

[ _ _

__<

t

a

j-

4t

t

(

l

-

:·

'2:

' l

l

.

l

-

(

J -r

J

2a

l

(T

H

f2

n)

I

-

,-

'

.

<

'

-2

-J

ia

a

=->

2

o.

[

(I +

-

-J

ia

-1

+

h

aJ

J

=8

<:

t l

a

a

=0

-

12

"

"

_

I

_ f

'l +

2

(

/2

,)

: l

&

,_

-

2

J

2

'

[·-

·

(

a+

h

)

1

-{

a

-

h)

=

"

"

&

<

'

- ~

>

- J

i a

a

+ l

+

2 a

'

~

o

Bu

t t

l1

i• q

oa

d

ra

tk

c

'q

ua

lto

n

wi

ll

ha

>e

t

wo

d

is

ti

(6

- 4

(

8)

(1

+

2a

2

1

>

0

7

2a

-

3

2

(I

+

2a

'

)

>0

7

2

a

-

n -

M

a

2

A

l

sa

'

-3

2:

>

0

a

-4

>

0

<t

2

>

4

~

a<

-

2

Ua>2

T

h

er

ef

or

e,

a

0'

(-

• -

2

) U

(

2,

""

) .

1

6.

Le

t

th

e

ra

diu

s

o

f h

e

ci

rc

le

be

r

_ J

'a

ke

x -

ax

is

a

lo

0

(

0,

0

) a

s

ce

nt

re

o

f

t

he

c

ir c

le

. T

h

er

etO

r

e,

o

f

A

an

d

C

ar

e

(

- r

,

0)

a

n

d

(r

, 0

) r

es

pe

c

tiv

el

y

N o

w

,

L

B

AC

=

jl.

L

8

0

C

2

~

T1

1e

re

for

o,

o

f

B

Elt

C (

r

c

os

2

13

. rs

in

A

nd

s

lo p

e

o

f A

D

I

S

H

li\

W

-

a

).

Le

t

(x .

y)

be

th

e

co

o

rd

in

a te

s

o

f th

e

p

oin

t D.

E

A

D

i

s

y

=t

a

n

(

)-

u )

(x

<-

r)

( ·

:

•l

op

e

(

fJ -

a

) a

nd

po

in

t

i

s (

-

r 0

))

N

o

w

,

Eq

u

at

io

n

o

f

B

C

; ;

rsm2jl-O

( )

Y-

x

-

r

r

oo

.>

21

}-

r

r

·2

s

in

jl

CO

S I}

y

=

--

x

-

r)

r(

-

2s

in

'

P

l

y

~

2

si

nP

c

os

p

x

-

r)

-

2s

in

2

P

y

=

-O

O

IP

x-

r)

T

o

ob

ta

in

t

he

co

o

rd

in a

te

o

f

D

so

l•

•

(

l)

s

im

u

lt

an

eo

us

ly



=> t a t l

j l - a )

x + r ) ~ -

o t j > ( x

- r )

=:.

l ] l - a

r t o l

a

~

1

=>

-<[tan

{D - a

)H o l

w

r [col

p

-ta

n (j' l- a

)]

~

rsin(

j3-a)

s ~ ]

[' 'p

,;n(

j3-o:)l

x

co ,U

l ,;_-)

_ _ ~ r

sin

j3 -cos (p

- a )

~

[sin (

il u)s

inP+eo

s(j} u

)cosj3]

cos j

3 -a ) ,;n

j3

r rO O

<

j

3 COS

(p • 0:

)

-Sln

jJ s

in

(fl_

-_a )I

'-

s

mjlco>

(j>-a)

x [cos

J> -

a )

cos

j3 + sin

(j

3 -

c )

sin

131

-r

[cos(j3

-a)oo

sj)

sinfls

m

k[cos

(j3 a

Pll=rfc

os(fl u I

fil

l

rw

s(2j3 a

)

x

-

c

osa

P

urring th

is value

in

(2 ), w

e get

-

v

=-cot

jJ' rcos(2

j: l-o:)

- r

L c

oso

J

::::

;, Y- co •

j3 r lcos

(2J>

a

)

co

sa I

""j\

cnsa

.

~ y=

- rco

s

p

l-2;in

2j3

; -a ,

;n a -2

1-t-a

l

sinp

co

sa

i

J

=>

,

_

_

s [ J [ 2 s

n ~ - s i n

Pll

·

.ini3

c

osa

~

- 2 r c o s

~ - > i n

(a

- jl)

i

co

' a

l h

ere fore.

coo rdin

ate ' of

D .m :

[

r en

• (

2jl

- )

_

2':_

cos_Jl

si

n

(«

-I. _)_ 'J

cos

a

cos a

T

hus, coo

mm:rtes

of

E

'

1

r e ~ > : ~

- ~ _ : < _

_ o o _ > _ a . _

r

co

s jhin

(a

--

j

lJ)

2cosa

cos

a

a

j

'

u

'

c o s ~ · c . o

s ( ~ a)

~ s i n (

· a )

r·

-

.

CO>

0: 0:

Smce .

.

J

t" - d, we ge

t

d

-

r'

r os pe

ns (IJ -a )+

] ] '

+

c'

-"'" D_<in

m

_

cos

u

L

cosa

=

.....>:;...- f

cos'

P

cos' (p

a ) + c

os

a

CO<-

<L

I '

I ' I

'

.

'

- , - · c o

cos

~ - O , -

S l n (

1

•

c

os· a

'

='

'

o

s· p •CO

O u +2

co '

~ t o >

"

Tilere

fare. ar

ea

of

the

circle

=JU'

'

os·

j;

1

c

os ·" 1

2cosCl

co;Bcoo

(j) "

The give

n circle

is,

ax

+2hxy

+i>_,

=I

Let

the po

in t P

~ i n g

(1)

bo (x

,J

1

),

le l 0

inclina

tion

of

l

ine throu

gh

P

w

hich

iH

ters<:<h

th

e

curve

>11 Q

01

1d R

.

Th

en equat

ion

of

line

til rough

I

r-

x 1 = y

- Yt =>"

co

sf l ,[r.

f.l

X= . r

1

+rcosf ,y=y,

+rsinO

fo r po

m tQ

and

R,

above

point m

ust lie o

n ( ll

~

a(x

1

+ rc

osf. l)

2

+2h

(x

1

~ r c - o s

I J ( y

, ' ' · '

- l>(y

1

'

" i1H

-=;,

(a

co

s'

f

.l

-

2h

sin f.lco

s 9

+ h'in

'

,

. 21

1

e

n< B. , m,

<m l

l

•

11<

·

1

11

h1·

- (axi

2h1, )",

+ lo

v; 1-

n

ll

is ~ u a d

a t i c in r

,

givi

ng mo

v

al"os

r JS

'{

""

:.

R ~ - -

.. x1+ 2

h 1y1

r h

,2 _

a cos

' e , 2hs

in e

c<>>

O

+

b,;,' o

He

ro, +

:-Ju, y,

+

b)\

- I

-,t 0.

a:; (.,

1

•

J.

)l l

, ,, ,

011 ( I)

Als

o,

a cos

'

0

+

2

h>in fl COl

fl + hoin

2

2 h s i

l c o s €

(b

,i

Jsin'O

=a + l (2hc

n ,fl

+ (

b -

)si

n

€1)

=

a +oin

fl. ;

.'4h

1

-

h

- a )

c o s 0

"

si'lL'c

b- 1

w

here.tan

fJ-

_

_

"

'

=a

+

• ~ l b ~ - - o - l

.1in e,in

€1 , 1

w

llich

wi

ll

inJq

;c nJcnl

ura. i

l'

4h

1 ~ - a J

~ o

:. Equati

on (1

i

1e

duces to

'

., '

y =

which

is

tl

circl

e.



·

Equat

ion' ot

;my

circle

C w it h

ce m

re

(

0,

,1 2) is gi

ven by

where

r ->

0

o

'

(x

01

· , (

y-,'2

1- ~

•

{

,·

,,

- y -

~ , ~

Y -

= r

-- 1 )

Let(x

1

, y

1

),

(x,,

Y z)

,(-< ,.

y

3

)be

thrcc

d isti

nctra

tio na

l

po

ints o

n l

). Sin

re a s tr

a igh t

line p

<m l.ll

elto x-

axis

mee

t;

a

oirol

e in a

t most

t w o

poin ts

, h e r

,1), y

2

or

Jl -

J',

Pu

tting

in

(1

).

we

get

S

uhtrac

iing

(2) fro

m (3 )

. we

obta in

,-

,f2q

,=O

wh

ere

q

=y

-y

Sub

tr,1cti

ng (2)

lm m

(4),

we ob

tain

p

, --

4, =0

whe1

e

p , = r i + y ~ - x , - ; 1 ,

q,-- > -F i

.( 2)

,(3)

- .. ( 4

)

N.,w,

11 , p

,,

q

1

, q ,a

re ration,

_]

num

bcro AI>

<>

eith

er

q

1

z0

orq

2

'>'0. If

q

1

lq

1

a

nd ifq

2

ct 0,

th

en ,h

=

p

,

I q

2

.

In an

y ca s

e .f

i

is

a r

ation

al num

be r.

This

i ; a c

ontra

dictio

n.

19 .

Le

t the p

oint P be

(2

r cos

e. 2r

>-in f

l)

W e h

ave

O

=

r,O

P

=2r

A•

A O

Af is

a r ig h t

a ngle

d tr ia

ng le ,

•

\

cos¢>

= l/2

=;.

$ = 7 t/

3

:. C o

ordin

ate. o

fA are

{r

co

s (G

- 1t I

3 .

rs i

n

(0

- 1t I J)

)

an

d th a

t of

B

are[

rcos

(

0 -rn

/ 3), rsin

(9 -r 1ti

3)]

tf p,

q is th

e cen

troid

of 1 'AB

,

the

n p = ~

[r e

o> 9

1

3) -r rro

s

(0 -r

1

t.

3) -> 2rc

rn;9

]

[

r

{co

s

(6

-

1

rl 3

+ cr

(6 + 1 /3

)}+

lr·o

os

0]

[[

·

-··

···

•

-'-•

-']

i rz

co s

~

3_uJ

>

_32

3

+

2rco

s e]

~

1

[r

{2

cos£1

cos 7

t/3} +

2rc<

Js6]

•

'

[r

·c

o s 9

rco

se

•

an d

q

=_

_

[n in

(o

- ::<

)., n

.i n (

e +

\2 r

s i

J

3

J)

= [r {>

in

l

;

s i n [ e

~

}

+ 2n

- [I,

..

:···:

:;

rl _sm

2

-

0

- -1

1

j

0.-

-\

.l

1

[r {2

>ine

co sn

l3 )+

2n in

6]

•

=

_ _ 1,. {s in

9} +2

cs in

O

J= >

·sine

l

p,q )=(roos6,nin9)

lies on

r

w h

ich is

cal lc

dC,_

20

.

'

-

c,

~ _ /

f

b :

:

V

~

Fr

om t1

gure i

t is cl

ear th

at tria

ngle

0/.

S

i

a rig

·ith

right

angle

m L.

Abu

UL=l

an d

S

=2

si<

l (L

L S O

) = =>

L

LSO

=

300

in

ce SA

1

=

.SA

2

, t1 SA

1

A

, is an

eq u

ilatera

l tri

T

he c

ircle w

ith c

entr<

at

C

is a

circl

e insc

r

,

\

S

A

1

A,_

The

refor

e, cen

tre

C, is c

entro

ld o

This.

C

1

di>id

e ' SM

in

the r

atio

2 : I

.

co

ordin

.-tes

of ( '

1

a

re (-4/3

,0)

a

nd

-C M

- l

/3



-. its equ

atLon io {x +

4 I

3)

1

<

i = (I

I 3

2

(1)

T h

e othc-r circle to

uche s the eq uila

tcrai triangle S

B

1

B,

e

xternally_ h.s

radiusr is give n

by ~ . w h

e r e

B

1

B,

=a. ut i\ = (a)(S.V)

= ~ a

' 2

2

Olld

s -a=-a -

a=a /2

2

l'hliS

,

r

=3

=oo

Coordinate

> ofC

2

are (4. 0)

s -a

·_

equation

of

i r d ~

with cen tre

m

c,

(x -4 )

+ y

=

3

2

-- .(2)

E q

ua t ion\ of

c om m

on tangents to c

ircle (I) and

mcle

C

.,

x • - 1 and

y=-T

-

2

- { x+2 )

J;

[T

1

nd

T,

1

Equation

of

com

m o n t

angent;

to

circlo

(2)

aud c

ircle

C

"

x= l and y - ± ~

( x , : : )

J2

7

1

andT,]

Two tangents co

m m o n

to

(l) an d

(2) are 1

1

and

l2 at

0

To lind the rema

inh>g

mo

traJls

-erse tangent>

to 1

)

and

(2),

we

t

lnd a potnl I whi

ch divides t l1e jo

int ofC

1

C

2

in

r a t 1 o ~

:r ,=

l 3 : 3 · : ~

Therefore,

coordinates

ofl are (-

4

I

5,

OJ

Equation

of an)' line

t h r o u I i

y

=

m

(x

+ 4 I 5). It il

l

touch

d ) if

~ - ~ ~ ~ ~ ~ 1

6 m

2

=25(1+m

2

t

3

9m

2

=25

'

= i

The

refore, these tan

gent< are

5 '

4)

v =

\ G < i l +

5

2 F.quati<>n

of

any tang

ent to

drclex

0

+

y

=r is

xcmO

+ys inO

-r

Suppose (

1)

i

tangent to

+

25

y

I

00

'

'

: _ ~ L -

1 a t ( x

,,2

25 4

1hen (1)

and

::::::c + m =1 ;ne identical

?_•

- _ ~ = ± =-

c

osO sinO

r

25co<9

4 s i n ~

=

>

x,

=

·Yt

~

'

'

The l

ine( ) meet the co ord

inates axes

in

A (r sec

e. 0) and

(0..- cosec 0). Let (h.

k) be 'nid

<>fA B.

k=rco;ece

'

h

erefore.

'

h= - - and 2

k =-'

>in e

As

(x

1

w

e

ge

l

cos 0

'

,•--

and

'

.l l =

2k

' '

j

lie'

on th

e ellipse

':_+ L

= 1

-

2

5 4

-(625 ~ I . . (

' ) ~ )

254

h i4k

~ ~ + - 2 _ , ,

4

h'

k

2

5k +4h =4h

k

There

fore, required lu

cuo is

4x

2

+25 y = 4x' J

• ' 0

2

.

2;(" + y

- ] J . } -

---

Lx -hr -J

:y+

v

=0

_;, h

x-y ) -y x -y

)= O

=>

(2x-y)(_,-y)

=0

---,

y

=2x

y=x

art the equations

o f strai[ h

l

pass ing thro

ugh origin.

N

ow, let the ang

le bet.,.,ccn

t ~ c

lin

es be

28

and th

y makes angl

e of 45" witll x-a

xis

Thcrcf<"

" '-"" ( 45° +

2 9)

= 2 (slope

of

he

line v -

~

~

tan45"-

tan29 _ =

2

1

tan

45'xtan2fl

l+tan20

a

n 29

(1

+tan

2fl)

-(1

-t•rllaJ

H

(l+tan20)+(1

ton29)

2 + 1)

1 tan 2fl

2

2

-

-

tan2fl=-

;

'

2

'

-

;

2 tan e

-

---

.

~ (2

t

an e 1

·3=1-ta

n

1

-tan'O

'

•n

9 + 6 t a n 0 - 1= 0



2

44

O

CA

wne=J ·.OA= -

O

A

tan

O

=

JO

Ji

O

J

(

- 3

O

J

(3

o

· -

- =

J

v

+ -

.1

J

(1 0

-

9 )

le t

th

e gi

ven

ci

rcl

es C

1

an

d

c,

hav

e

ce

ntr

.,.

o

an

d

0

2

a

nd

ra

dii

,. ,

a

nJ

r re

sp

oc

liv

dj.

L

et

rh e

'a r

i b

lc

cir

cle

C

to

uc

h i

ng

C

1

i

ntc

ma

ll}

, C

,

c'

tc r

na

ll y

h

a'e

a

r&

liu>

ra

nd

cen

tre

at 0

.

N

ow

,

\

·-·

--<

..

.

0

0

2

~ r

r

,

an

d

1

r ·

· r

1

-

00

~

r

1 r

2

wh

1ch

i

gr

ea t

.,. th

an

0

1

0

2

a

s

0

1

0 , <

r

1

r

0

(

C

, lie

s

in ,

.dc

C

1

)

=

> iO

<U

< o

f()

is

an

el

hp>

e

W

iih

f

oc

i 0

1

a

nd

0

,.

l

te r

na

te

su

lu

tiu

n

:

l.e

t

cq

uat

iOl

l.> o

f

C

1

b

e

x <·

y

an

d

oi

c

,

be

.

.

n

-

a l

·

+

(y

·

b

r

le

i

cem

re

C

b

t:

(h

. k

) an

d r

adi

uo

r

th

en

by

tJ,c

g

ive

n

\i

(h

a

l

2

+(

k-

b

/

=r

+r

, a

nd

,lh

2

.

k

2

- ~

r

SS

ER

TIO

N N

D

RE

S

ON

L.

S

in c

e th

e

ar

e p

erp

en

dic

ula

r.

S

o.

l

<>e

u'

of p

er p

end

ic

ula

r

tru

1g

em

s

to

cir

elc

T

CH

T

HE

CO

LU

MN

l .

(A )

W

he

n

tw

o

cir

cle

s

arc

i

nte

rse

cti

ng

: t

hey

h

a,

·e

a

co

m1

non

n

orm

al

an

d

C

<Jm

mo

n tan

ge

n l

2

5.

rep

re

sen

t.

an

e

llip

><e

wh

ose

li

,ci

ar

e a

t

(0

. 0)

.

I fle

eq

ua

tio

n o

f

cir

cle

ha

vin

g

t

on

ge

nt

2

c +

3

(

I, - I

=>

( r

· l l

+

(y

+

l)

w

htc

h ;,

Ol1

1w

go

nal

(0

1l:

e c

ird

e h

avi

ng

en

d

mm

et

'

(0,

-

I)

an

d (

-

2,

3

)

=

>

.

r (x

+

2 )

(y

o

i) (

J ·

·3

)=

0

=>

2i

--·

·J

=

>

Jn

. .f

rom

e<

J< '-

Ii<>

n (

J)

eq

ua

ti o

n of

c

irc

le .

.

>:

2 y - 1 0 x - 5 y I ~

O

t

he

ci

rc l

es

w i

th

ra

dii

3. 4

an

d 5

t

ou

ch

ext

ern

al]

}·

and

P

is

the

po

in

t o

f i

nte

«e

ci i

on

ot

=>

1

ts

in ccntr c

ofi' .C 1

C

,c :.

T

hu

s d i

,ta

nc

e

of

p

oin

t

P f

rom

t

he

pu

im

s o

t c

r

ad

i s

(r

) ,,

f il.

C

1

C

2

C

1

.

c

i

o .,

b)

{s

c)

he

re

2s

=7

+

&+

9

H

cn

oe

,

y

-1

(, 9

is

a

dir

cc

w r

c

ir c

le

e4u

at

2

•

0

0

X

•}

-

> .

>o .

(l

I W

h

en

on

e c

ir c

lo l

ies

in,

de

of

o

th e

r lh

en

.

U

>Tl

lmo

n n

orm

al

bu

t n

o

c<

lmH

\O

ll wn

ge

nl.

(D

) 1

w

o b

rru

Jcb

e

h

H

erb

ol

a ha

ve

a c

om

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