The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a, is

Correct answer: (D) acot ((π)/(2 n) ). tan ((π)/n )=a/(2 r) ; sin ((π)/n )=a/(2 R) r+R=a/2 [cot (π)/n+cosec (π)/n ] ⇒ r+R=a/2 · cot ((π)/(2 n) )

Q104 / 225 · JEE Main 2003 · Mathematics

The sum of the radii of inscribed and circumscribed circles for an $\mathrm{n}$ sided regular polygon of side a, is

A
$\frac{a}{4} \cot \left(\frac{\pi}{2 n}\right)$
B
$a \cot \left(\frac{\pi}{n}\right)$
C
$\frac{a}{2} \cot \left(\frac{\pi}{2 n}\right)$
Correct answer
D
$\operatorname{acot}\left(\frac{\pi}{2 n}\right)$

Solution

$\tan \left(\frac{\pi}{n}\right)=\frac{a}{2 r} ; \sin \left(\frac{\pi}{n}\right)=\frac{a}{2 R}$ $r+R=\frac{a}{2}\left[\cot \frac{\pi}{n}+\operatorname{cosec} \frac{\pi}{n}\right] \Rightarrow r+R=\frac{a}{2} \cdot \cot \left(\frac{\pi}{2 n}\right)$

Hence, the correct answer is option: 'D'.

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