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Question

If the radical axis of the circles x2+y2+2gx+2fy+c=0 and 2x2+2y2+3x+8y+2c=0 touches the circle x2+y2+2x+2y+1=0, Then

A
g=34 or f=2
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B
g=14 or f=2
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C
g=34 or f=1
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D
g=34 or f=2
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Solution

The correct option is D g=34 or f=2
The radical axis of the given circles x2+y2+2gx+2fy+c=0 and x2+y2+(32)x+4y+c=0, is (2g32)x+(2f4)y=0 or (4g3)x+4(f2)y=0 (1)

This radical axis (1) touches the circles x2+y2+2x+2y+1=0, ...(2)

If the length of from centre (1,1) on the line (1)= radius of circle (2) i.e.,
(4g3)(1)+4(f2)(1)[(4g3)2+16(f2)2]=±(1+11)

[(4g3)+4(f2)]2=(4g3)2+16(f2)2
8(4g3)(f2)=0
g=34 or f=2

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