Question

A circle S of radius 'a' is the director circle of another circle $S_1$. $S_1$ is the director circle of circle $S_2$ and so on. If the sum of the radii of all these circles is $2$, then the value of a is

• A. $2+\sqrt{2}$
• B. $2-\sqrt{2}$
• C. $2-1/\sqrt{2}$
• D. $2+1/\sqrt{2}$

Answer 1

The correct option is B $2-\sqrt{2}$

Director circle is the locus point of intersection of tangent

If r is the radius of circle then $\sqrt{2}r$ is the radius of the director circle

Given radius of $S=a$

Then radius of $S_1=\frac{a}{\sqrt{2}}$

$S_2=\frac{\frac{a}{\sqrt{2}}}{\sqrt{2}}$

$S_n=\frac{S_{n-1}}{\sqrt{2}}$

Sum = $a+\frac{a}{\sqrt{2}}+\frac{a}{2}+\frac{a}{2\sqrt{2}}\dots \dots$

Sum of infinite GP = $\frac{a}{1–r}=\frac{a}{1-\frac{1}{2}}=2$

$2=\frac{\sqrt{2}a}{\sqrt{2}-1}$

$a=2-\sqrt{2}$