GATE 2020 AG – Question 8

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

A maximum-area rectangle PQRS is inscribed in a circle of radius a. Find the shaded area outside the rectangle and inside the circle.

$\pi a^2-a^2$
$\pi a^2-\sqrt2a^2$
$\pi a^2-2a^2$
$\pi a^2-3a^2$

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Show answer and explanation

Correct answer: (C) $\pi a^2-2a^2$

Explanation

An inscribed rectangle has diagonal 2a, so its sides satisfy x²+y²=4a². Since 2xy≤x²+y², its largest area is xy=2a² when x=y. Subtract from circle area πa² to obtain (π−2)a², C.

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