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

Option: 1
$\mathrm{2+\sqrt{2}}$

Option: 2
$2-\sqrt{2}$

Option: 3
$2-\frac{1}{\sqrt{2}}$

Option: 4
$2+\frac{1}{\sqrt{2}}$

Answers (1)

best_answer

Radius of circle $\mathrm{S=a}$

Radius of circle $\mathrm{S_1=\frac{a}{\sqrt{2}}}$
Radius of circle $\mathrm{S_2=\frac{a}{(\sqrt{2})^2}}$ and so on.

$\mathrm{\therefore }$ sum of radii $\mathrm{=a+\frac{a}{\sqrt{2}}+\frac{a}{2}+\frac{a}{2 \sqrt{2}}+\ldots . . }$

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

$\mathrm{ \Rightarrow a=2 \left(1-\frac{1}{\sqrt{2}}\right)=2-\sqrt{2} \\ }$

Hence option 2 is correct.

Posted by

Sanket Gandhi

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