7. 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 \(e\).
- 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 \(A ^ { \prime }\) are the points of intersection of \(l\) with the asymptotes of \(H\). - Determine, in terms of \(e\), the length of the line segment \(A A ^ { \prime }\).
The point \(F\) is the focus of \(H\) for which \(x < 0\)
Given that the area of triangle \(A F A ^ { \prime }\) is \(\frac { 164 } { 3 }\) - show that \(a\) is a solution of the equation
$$30 a ^ { 3 } - 164 a ^ { 2 } + 375 a - 4100 = 0$$
- Hence, using algebra and making your reasoning clear, show that the only possible value of \(a\) is \(\frac { 20 } { 3 }\)