Q47 / 90 · JEE Main 2010 · Mathematics

For a regular polygon, let $\mathrm{r}$ and $\mathrm{R}$ be the radii of the inscribed and the circumscribed circles. A false statement among the following is

  • A
    There is a regular polygon with $\frac{r}{R}=\frac{1}{\sqrt{2}}$
  • Correct answer
    B
    There is a regular polygon with $\frac{r}{R}=\frac{2}{3}$
  • C
    There is a regular polygon with $\frac{r}{R}=\frac{\sqrt{3}}{2}$
  • D
    There is a regular polygon with $\frac{r}{R}=\frac{1}{2}$

Solution

$$ \mathrm{r}=\frac{\mathrm{a}}{2} \cot \frac{\pi}{\mathrm{n}} $$ 'a' is side of polygon. $$ \begin{aligned} & R=\frac{a}{2} \operatorname{cosec} \frac{\pi}{n} \\ & \frac{r}{R}=\frac{\cot \frac{\pi}{n}}{\operatorname{cosec} \frac{\pi}{n}}=\cos \frac{\pi}{n} \\ & \cos \frac{\pi}{n} \neq \frac{2}{3} \text { for any } n \in N \end{aligned} $$

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

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