The Orthoptic Nerve of Some Ellipses

Since $e = \frac{7}{25} = \frac{c}{a}$ , $c = 7k$ and $a = 25k$ , and since $a^2 - b^2 = c^2$ in an ellipse, $b = 24k$ .

The area of an ellipse is $A = \pi ab$ , so $A_E = 600 \pi k^2$ .

The set of points for which two tangents of any curve meet at a right angle is an orthoptic , and an orthoptic for any ellipse is a circle with a radius of $\sqrt{a^2 + b^2}$ , and therefore an area of $A = \pi (a^2 + b^2)$ , so $A_F = 1201 \pi k^2$ .

Therefore, $\frac{A_E}{A_F} = \frac{600 \pi k^2}{1201 \pi k^2} = \frac{600}{1201}$ , so $p = 600$ , $q = 1201$ , and $p + q = \boxed{1801}$ .