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Question

A circle x2+y2+4x22y+c=0 is the director circle of circle S1 and S1 is the director circle of circle S2 and so on. If the sum of radii of all these circles is 2, then the value of c=k2, where value of k is

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Solution

The radius of the given circle x2+y2+4x22y+c=0 is 4+2c=6c=a (let).
Now, radius of circle S1=a2
Radius of circle S2=a2
and so on.
Also, a+a2+a2++=2 (Given)
Then a⎜ ⎜ ⎜ ⎜1112⎟ ⎟ ⎟ ⎟=2
or a221=2
or a=22=6c
or 4+242=6c
or c=42

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