Problems

01

Biggest rectangle inscribed in ellipse

A rectangle is inscribed in the ellipse given by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. The four points of this rectangle are given in coordinates by the four possibilities for $(\pm x,\pm y)$. Using the method of Lagrange multipliers, find the largest possible area of such a rectangle in terms of the constants $a,b$.