Question

A circle $x^2+y^2+4x-2\sqrt{2}y+c=0$ is the director circle of circle $S_1$ and $S_1$ is the director circle of circle $S_2$ and so on. If the sum of radii of all these circles is $2$, then the value of $c=k\sqrt{2}$, where value of $k$ is

Answer 1

The radius of the given circle $x^2+y^2+4x-2\sqrt{2y}+c=0$ is $\sqrt{4+2-c}=\sqrt{6-c}=a$ (let).
Now, radius of circle $S_1=\frac{a}{\sqrt{2}}$
Radius of circle $S_2=\frac{a}{2}$
and so on.
Also, $a+\frac{a}{\sqrt{2}}+\frac{a}{2}+\dots \dots +\infty =2$ (Given)
Then $a\left(\frac{1}{1-\frac{1}{\sqrt{2}}}\right)=2$
or $\frac{a\sqrt{2}}{\sqrt{2}-1}=2$
or $a=2-\sqrt{2}=\sqrt{6-c}$
or $4+2-4\sqrt{2}=6-c$
or $c=4\sqrt{2}$