Question

The sides of a rectangle of the greatest area which can be inscribed into an ellipse x^{​​​​​​2}/25 + y^{​​​​​​2}/9 = 1 are

Answer 1

The vertices of any rectangle inscribed in an ellipse is given by (±acos(θ),±bsin(θ)) The area of the rectangle is given by A(θ)=4abcos(θ)sin(θ)=2absin(2θ) Hence, the maximum is whensin(2θ)=1. Hence, the maximum area is when2θ=π/2 i.e.θ=π/4. The maximum area is A=2ab [ for ellipse x^2/a^2 +y^2/b^2=1] hence comparing with given equation a=5 b=3 so area A =2×5×3 A=30