A circle with centre O is shown in the figure. A rectangle PQRS of maximum possible area is inscribed in the circle. If the radius of the circle is a, then the area of the shaded portion (circle minus rectangle) is _______. A. pi*a^2 - a^2 B. pi*a^2 - sqrt(2)*a^2 C. pi*a^2 - 2*a^2 D. pi*a^2 - 3*a^2

GATE 2020 · General Aptitude · Circle · medium

Answer: The area of the shaded portion is pi*a^2 - 2*a^2.

1. Identify the maximum-area inscribed rectangle: Here r = a, so diagonal of rectangle = 2a. For a square with diagonal d, side = d / sqrt(2) = 2a / sqrt(2) = a*sqrt(2).
2. Compute the area of the inscribed square: Area of square = (a*sqrt(2)) x (a*sqrt(2)) = 2a^2.
3. Compute the area of the circle: Area of circle = pi * a^2.
4. Compute the shaded area: Shaded area = pi*a^2 - 2a^2.