JEE Main · 2005 · 08 May 2005 · Offline · Application of Derivatives

Area of the greatest rectangle that can be inscribed in the ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

Answer: (A)

Solution

Take a general vertex of an inscribed rectangle as

(a cos θ, b sin θ)

; by symmetry the rectangle then has sides

2 a cos θ

and

2 b sin θ

. Its area is

(2 a cos θ) (2 b sin θ) = 2 ab sin 2 θ

. Because

sin 2 θ

has maximum value

1

, attained at

θ = \frac{π}{4}

, the greatest area of a rectangle inscribed in the ellipse is

2 ab

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option