Maximum Parallelogram Area within an Ellipse

Calculus Level 4

Given the ellipse:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

Two points \( (x_1, y_1) \) and \( (x_2, y_2) \) are chosen on it. A parallelogram is drawn using the points \( (x_1, y_1) \) and \( (x_2, y_2) \) and \( (-x_1, -y_1) \) and \( (-x_2, -y_2) \). What is the maximum area of this parallelogram ?

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