Question

Find the area of the greatest rectangle that can be inscribed in the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1.`

Answer 1

Reactangle can be inscribed in the ellipse with its sides parallel ot axes of theellipse as shown in the following figure.

Let `P-=(a cos theta, b sin theta)`
`:.` Area o reactangle PQRS
`=(2 a cos theta) (2b sin theta)`
`= 2ab sin 2 theta`
This is maximum when `sin 2 theta=1`
Henece, maximum area `=2ab(1)=2ab`