Let G be a circle of radius R > 0 . Let G _ 1 , G _ 2 , , G _ n be...

Plotting the diagram of given condition we have,

Now each radius of smaller circle will make $\dfrac{\pi}{n}$ angle at centre of bigger circle,

Now applying the sine rule, we get

$\dfrac{\sin \left(\dfrac{\pi}{n}\right)}{sin\dfrac{\pi}{2}} = \dfrac{r}{R + r}$

$⇒ \dfrac{R}{r} + 1 = cosec \left(\dfrac{\pi}{n}\right)$

$⇒ R = r\left[cosec \left(\dfrac{\pi}{n}\right)\right]\left[- 1\right]$

Now checking all options we get,

(A) $n = 4, R = r\left(\sqrt{2} - 1\right)$

(B) $n = 5, R = r\left(cosec \dfrac{\pi}{5} - 1\right)$

$⇒ R < r\left(cosec \dfrac{\pi}{6} - 1\right) ⇒ R < r$

(C) $n = 8, R = r\left[cosec \left(\dfrac{\pi}{8}\right) - 1\right]$

$⇒ R > r\left(cosec \left(\dfrac{\pi}{4}\right) - 1\right)$

$⇒ R > r\left(\sqrt{2} - 1\right)$

(D) $n = 12, R = r\left[cosec \left(\dfrac{\pi}{12}\right) - 1\right]$

$⇒ R = \left[\sqrt{2}\left(\sqrt{3} + 1\right) - 1\right]r$

$⇒ R < \sqrt{2}\left(\sqrt{3} + 1\right)r$