Problem

A hyperbola $H$ has equation

$$\frac{x^2}{a^2} - \frac{y^2}{25} = 1$$

where $a$ is a positive constant.

The eccentricity of $H$ is $e$.

(a) Determine an expression for $e^2$ in terms of $a$.

The line $l$ is the directrix of $H$ for which $x > 0$. The points $A$ and $A'$ are the points of intersection of $l$ with the asymptotes of $H$.

(b) Determine, in terms of $e$, the length of the line segment $AA'$.

The point $F$ is the focus of $H$ for which $x < 0$. Given that the area of triangle $AFA'$ is $\frac{164}{3}$,

(c) show that $a$ is a solution of the equation

$$30a^3 - 164a^2 + 375a - 4100 = 0.$$

(d) Hence, using algebra and making your reasoning clear, show that the only possible value of $a$ is

$$\frac{20}{3}.$$