MathematicsFind the area of the greatest rectangle that can be inscribed in a ellipse x2+4y2=4.

Find the area of the greatest rectangle that can be inscribed in a ellipse $x^{2} + 4 y^{2} = 4$ .

Given ellipse is

$\begin{array}{l} x^{2} + 4 y^{2} = 4 \\ \Rightarrow \frac{x^{2}}{4} + \frac{y^{2}}{1} = 1 \end{array}$

Let $A$ be the area of the rectangle.

So, $A = (4 \cos θ) (2 \sin θ) = 4 \sin (2 θ)$

Thus, the greatest area of the rectangle is $4$ .

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