CAT 1996 — QA Question 21

Let radius of the semicircle be R and radius of the circle be r.
Let P be the centre of semicircle and Q be the centre of the circle.
Draw QS parallel to BC.
Now, ΔPQS ~ ΔPBC

$$\begin{gathered} \therefore \frac{PQ}{PB}=\frac{QS}{BC} \\ \Rightarrow \frac{R+r}{\sqrt{2}R}=\frac{R-r}{R} \\ \Rightarrow R+r=\sqrt{2}R-\sqrt{2}r \\ \Rightarrow r(1+\sqrt{2})=R(\sqrt{2}-1) \end{gathered}$$

$\Rightarrow r=R\frac{(\sqrt{2}-1)}{(\sqrt{2}+1)}\times \frac{(\sqrt{2}-1)}{(\sqrt{2}-1)}$

$\Rightarrow r=R{(\sqrt{2}-1)}^2$

Required Ratio = $\frac{\pi r^2}{\pi R^2}\times 2$

$=\frac{\pi R^2{(\sqrt{2}-1)}^4\times 2}{\pi R^2}$

$=2{(\sqrt{2}-1)}^4:1$