Question

Using integration find the area of the largest rectangle that can be inscribed in an ellipse

Answer 1

The vertices of any rectangle inscribed in an ellipse is given by

(±acos(θ),±bsin(θ))(±acos⁡(θ),±bsin⁡(θ))

The area of the rectangle is given by

A(θ)=4abcos(θ)sin(θ)=2absin(2θ)A(θ)=4abcos⁡(θ)sin⁡(θ)=2absin⁡(2θ)

Hence, the maximum is when sin(2θ)=1sin⁡(2θ)=1. Hence, the maximum area is when 2θ=π22θ=π2 i.e. θ=π4θ=π4. The maximum area is

A=2ab

Answer 2

I have given the answer earlier