JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Application of Derivatives

The triangle of maximum area that can be inscribed in a given circle of radius '

r

' is :

Answer: (B)

Solution

For a triangle inscribed in a circle of radius

r

the sides are

a = 2 r sin A

b = 2 r sin B

c = 2 r sin C

, so its area is

Δ = \frac{1}{2} ab sin C = 2 r^{2} sin A sin B sin C

The angles are constrained only by

A + B + C = π

, and

lo g sin

is concave on

(0, π)

, so

sin A sin B sin C

is largest when the three angles are equal. The maximising triangle is therefore equilateral with

A = B = C = \frac{π}{3}

, and each side measures

2 r sin \frac{π}{3} = 3 r

The triangle of maximum area is the equilateral triangle with each side of length

3 r

JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Single correct option