Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, if the area of the rectangle formed by the four points of intersection of the ellipse with the lines $x = \pm ae$ and $y = \pm \frac{b^2}{a}$ is defined, what is the ratio of this area to the area of the ellipse when $e = \frac{\sqrt{3}}{2}$?