The rectangle with the largest area that can be enclosed in a circle of radius $r$ is of course a square. Its sides have length

$s=$ and its area is $A=$

Consider now the ellipse defined by

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

The rectangle with the largest area that can be inscribed in that ellipse has a horizontal side of length $,$ a vertical side of length $,$ and an area

Hint: You can solve the ellipse problem by cranking the handle and proceeding as we did for several similar geometric problems. Or you can look at the results for the circle and make an inspired guess.