A) \[a\cot \,\left( \frac{\pi }{v} \right)\]
B) \[\frac{a}{2}\cot \left( \frac{\pi }{2n} \right)\]
C) \[a\cot \,\left( \frac{\pi }{2n} \right)\]
D) \[\frac{a}{4}\cot \left( \frac{\pi }{2n} \right)\]
Correct Answer: B
Solution :
AB = a (let) \[ON\bot AB\] and AN = BN In \[\Delta AON\], \[\tan \frac{\pi }{n}=\frac{AN}{ON}\] \[ON=AN\cot \frac{\pi }{n}\] \[=\frac{a}{2}\cot \frac{\pi }{n}\] .... (i) and \[\sin \frac{\pi }{n}=\frac{AN}{OA}\] \[OA=AN\cos ec\frac{\pi }{n}=\frac{a}{2}\cos ec\frac{\pi }{n}\] ... (ii) Sum of the radii = ON + OA \[=\frac{a}{2}\cot \frac{\pi }{n}+\frac{a}{2}\cos ec\frac{\pi }{n}\] \[=\frac{a}{2}\left[ \frac{\cos \frac{\pi }{n}}{\sin \frac{\pi }{n}}+\frac{1}{\sin \frac{\pi }{n}} \right]\] [from Eqs. (i) and (ii)] \[=\frac{a}{2}\left[ \frac{1+\cos \frac{\pi }{n}}{\sin \frac{\pi }{n}} \right]=\frac{a}{2}\left[ \frac{1+2{{\cos }^{2}}\frac{\pi }{2\,n}-1}{2\,\sin \frac{\pi }{2\,n}\cos \frac{\pi }{2\,n}} \right]\] \[=\frac{a}{2}\cos \frac{\pi }{2\,n}\]You need to login to perform this action.
You will be redirected in
3 sec