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Question

For circles (x3)2+(y2)2=9 and (x+5)2+(y+6)2=9 which among the following are tangent equations

A
y+2=163237(x+1)
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B
y+2=16+3237(x+1)
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C
y+1=x+32
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D
y+1=x32
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Solution

The correct options are
A y+2=163237(x+1)
B y+2=16+3237(x+1)
C y+1=x+32
D y+1=x32

C1(3,2) r1=3

C2(5,6) r2=3


Internal point of simlitude (I) divide C1 and C2 in 1:1 ratio internally
​​​​​​so, I(1,2)
Let the equation of tangents passing through I be

y+2=m(x+1)
mxy+m2=0

Perpendicular distance from C1 to the tangent = 3
4m41+m2=316(m1)2=9(m2+1)7m232m+7=0m=16±3237
So, equation of Transverse common tangents are
y+2=16±3237(x+1)
Radius of both circles are same.

Direct Common tangent are parallel to C1C2, so, D.C.T. equation will be of the form
y=x+λ (mC1C2=1)
Perpendicular distance = radius
1+λ2=3

λ+1=±32
λ=±321

So, equation of D.C.T. are y=x+321 and y=x321

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