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Question

Find the equations of transverse common tangents for two circles

x2 + y2 + 6x − 2y + 1 =0 , x2 + y2 − 2x − 6y + 9 = 0


A

35x2 + 12xy 18x = 0

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B

35y2 + 12xy 18y = 0

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C

3x2 4xy + 16y 12x = 0

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D

3y2 4xy + 16x 12y = 0

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Solution

The correct option is A

35x2 + 12xy 18x = 0


Given circles,

x2+y2+6x2y+1=0 - - - - - - (1)

x2+y22x6y+9=0 - - - - - - (2)

Let the circle be c1 & c2 and radii r1 & r2 of circle (1) and (2) respectively

c1(3, 1)

c2(1,3)

c1c2=16+4=25=4.47

r1=g2+f2c=9+11=3

r2=g2+f2c=1+99=1

c1c2>r1+r2

Both the circle lies outside each other.

Let the intersection point of tarnsverse common tangent be P. P divides c1 & c2 internelly in the ratioo of

r1:r2 here ,in the ratio of 3:1.Co-ordinate of P(x,y)=(m1x2+m2x1m1+m2,m1y2+m2y1m1+m2)

=(3×1+1×133+4,3×3+1×13+1)

=(0,52)

Equation of transvers common tangent or pair of tangent from the point P is

T2=SS1 - - - - -(3)

T=xx1+yy1+3(x+x1)(y+y1)+1

=x × 0y52+3(x+0)(y+52)+1

=52y+3xy52+1=32y+3x32

32(y+6x1)

s1=(0)2+(52)2+6(0)2(52)+1

=254102+1=2520+44=94

Substituting values of T & s1 in equation (3)

94(y+6x1)2=(x2+y2+6x2y+1)×94

35x2+12xy18x=0

Equation of transverse common tangents

35x2+12xy18x=0


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