CAT 2023 Slot 3 QA Question 17: A rectangle with the largest possible area is drawn inside a semicircle of radius 2 cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is

Question

A rectangle with the largest possible area is drawn inside a semicircle of radius 2 cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is

Accepted Answer

Let half the base of the rectangle be x and its height be y. The top corner lies on the semicircle of radius 2.

$x^2 + y^2 = 4$

The rectangle has base $2x$ and height y, so its area is

$A = 2xy$

For fixed $x^2 + y^2$, the product xy is maximum when $x = y$.

$x = y = \sqrt{2}$

Thus the rectangl