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Question

Find the area of the greatest rectangle that can be inscribed in a ellipes x2a2+y2b2=1.

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Solution

The vertices of any rectangle inscribed in an ellipse is given by (±acosθ,±bsinθ)

The area of the rectangle is given by A(θ)=4abcosθsinθ=2ab.sin(2θ)

Hence, the maximum is when sin2θ=1. Hence, the maximum area is when 2θ=π2 i.e. θ=π4.

The maximum area is A=2ab.


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