GATE 2020 EC GA Q.8: solution and answer

GATE 2020 · ECGA Q.8 · 2 marks · MCQ

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 is ______.

GATE 2020 EC GA question 8: A circle with centre O (marked with a small cross, labelled O just above it). — Figure from the GATE 2020 EC paper. It is part of the question.

(A)
$π a^{2} - a^{2}$
(B)
$π a^{2} - 2 a^{2}$
(C)
$π a^{2} - 2 a^{2}$
(D)
$π a^{2} - 3 a^{2}$

Answer and solution

ANSWER

The shaded area is $π a^{2} - 2 a^{2}$ , option (C).

Official answer: CSource: Official answer key, GATE 2020 EC (IIT Delhi)

Shaded area = area of circle $-$ area of rectangle $= π a^{2} - area (P QR S)$ .
A rectangle inscribed in a circle has its diagonal as a diameter: the right angle at $P$ is an angle in a semicircle, so the chord $S Q$ it subtends is a diameter. Hence $S Q = 2 a$ , and the segment $S O = a$ drawn in the figure is half of this diagonal.
Let the sides be $x$ and $y$ . Pythagoras in triangle $S P Q$ gives $x^{2} + y^{2} = (2 a)^{2} = 4 a^{2}$ .
Since $(x - y)^{2} \geq 0$ , we have $2 x y \leq x^{2} + y^{2}$ , so $x y \leq \frac{x ^{2} + y ^{2}}{2} = \frac{4 a ^{2}}{2} = 2 a^{2}$ , with equality only when $x = y$ .
So the rectangle of maximum area is a square of side $x = y = 2 a$ , and its area is $(2 a)^{2} = 2 a^{2}$ .
Shaded area $= π a^{2} - 2 a^{2}$ , which is option (C).

KEY IDEA

A rectangle inscribed in a circle always has a diameter ( $2 a$ ) as its diagonal. For a fixed diagonal, $x^{2} + y^{2} = 4 a^{2}$ is fixed, and the product $x y$ is largest when $x = y$ . So the maximum-area inscribed rectangle is a square of area $2 a^{2}$ , and the shaded area is $π a^{2} - 2 a^{2}$ .

WATCH OUT FOR

Taking the measurements from the figure. The rectangle is drawn wider than it is tall, but "maximum possible area" forces a square, so the drawing is not to scale. Another common slip is to treat $a$ as the half-side or as the whole diagonal instead of the radius. That gives a square area of $a^{2}$ or $2 a^{2}$ and leads to options (A) or (B).

From the GATE 2020 EC paper. Every question of this paper · Download the full paper · All 22 Geometry questions, with notes

GATE 2020 EC GA Q.8: solution and answer

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