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Question

If two circles with centres at (a,0) and (−a,0) having radii b and c units respectively such that a>b>c. Then the point of contacts of common tangents to these two circles will always lie on

A
x2+y2=a2±bc
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B
x2y2=a2±bc
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C
x2+2y2=a2±bc
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D
2x2+y2=a2±bc
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Solution

The correct option is A x2+y2=a2±bc
The equation of the given circles be
C1:x2+y22ax+a2b2=0C2:x2+y2+2ax+a2c2=0
Let P(h,k) be a point of contact of common tangents to these circles such that it lies on C1. Then
h2+k22ah+a2b2=0(1)
Now equation of tangent at P on C1 will be
hx+kya(x+h)+a2b2=0x(ha)+kyah+a2b2=0
This will also touch the circle C2
|a(ha)ah+a2b2|(ha)2+k2=c|2a22ahb2|b=c2ah(2a2b2)=±bc2ah(a2b2)=a2±bch2+k2=a2±bc (using (1))
Hence locus will be
x2+y2=a2±bc

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