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 $2$ units and the value of $c$ is $4\sqrt{k},$ then the value of $k$ is

Answer 1

Radius of given circle $=\sqrt{4+2-c}=\sqrt{6-c}=a$ (say)
Radius of circle $S_1=\frac{a}{\sqrt{2}}$
Radius of circle $S_2=\frac{a}{2}$ and so on.
$\therefore a+\frac{a}{\sqrt{2}}+\frac{a}{2}+\cdots \infty =2$
$\Rightarrow a\left[\frac{1}{1-\frac{1}{\sqrt{2}}}\right]=2\Rightarrow a\left(\frac{\sqrt{2}}{\sqrt{2}-1}\right)=2$
$\Rightarrow a=2-\sqrt{2}\Rightarrow \sqrt{6-c}=2-\sqrt{2}$
$\Rightarrow 6-c=4+2-4\sqrt{2}\Rightarrow c=4\sqrt{2}$
Hence, $k=2$