NIMCET 2016 — Mathematics PYQ

Where $2a$ is the length of the major axis and $2b$ is the length of the minor axis.  

Calculation: 
Let the coordinate points of the edges of the rectangle be (see diagram)  

$(a\cos \theta ,b\sin \theta )$, $(-a\cos \theta ,b\sin \theta )$, $(-a\cos \theta ,-b\sin \theta )$, and $(a\cos \theta ,-b\sin \theta )$  

The length of the rectangle = 2a cosθ
The breadth of the rectangle = 2b sinθ
Area of the rectangle = 2a cosθ × 2b sinθ
⇒ Area of the rectangle = 4ab cosθ sinθ = 2ab sin2θ (∵ 2 sinθ cosθ = sin2θ)
To maximize area the value of sin2θ should be maximum, i.e., sin2θ = 1
Maximum area of the rectangle in the ellipse = 2ab