A circle is circumscribed around a square as shown in the figure. The area of one of the four shaded portions is equal to $$\frac{4}{7}$$. The radius of the circle is :

Solution (By Examveda Team)

Let the radius of the circle be r and the side of the square be a
Then,
Diagonal of the square = 2r
⇒ $$\sqrt 2 $$ a = 2r
⇒ a = $$\frac{2}{{\sqrt 2 }}$$ r
⇒ a = $$\sqrt 2 $$ r
Area of one shaded portion :
$$\eqalign{ & = \frac{1}{4}\left[ {\pi {r^2} - {{\left( {\sqrt 2 r} \right)}^2}} \right] \cr & = \frac{1}{4}\left( {\pi - 2} \right){r^2} \cr & \therefore \frac{1}{4}\left( {\pi - 2} \right){r^2} = \frac{4}{7} \cr & \Rightarrow \left( {\frac{{22}}{7} - 2} \right){r^2} = \frac{{16}}{7} \cr & \Rightarrow {r^2} = \frac{{16}}{7} \times \frac{7}{8} \times 2 \cr & \Rightarrow r = \sqrt 2 \cr} $$