8. The hyperbola \(H\) has equation
$$4 x ^ { 2 } - y ^ { 2 } = 4$$
- Write down the equations of the asymptotes of \(H\).
- Find the coordinates of the foci of \(H\).
The point \(P ( \sec \theta , 2 \tan \theta )\) lies on \(H\).
- Using calculus, show that the equation of the tangent to \(H\) at the point \(P\) is
$$y \tan \theta = 2 x \sec \theta - 2$$
The point \(V ( - 1,0 )\) and the point \(W ( 1,0 )\) both lie on \(H\).
The point \(Q ( \sec \theta , - 2 \tan \theta )\) also lies on \(H\).
Given that \(P , Q , V\) and \(W\) are distinct points on \(H\) and that the lines \(V P\) and \(W Q\) intersect at the point \(S\), - show that, as \(\theta\) varies, \(S\) lies on an ellipse with equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$
where \(a\) and \(b\) are integers to be found.
\includegraphics[max width=\textwidth, alt={}]{d7a92540-e36c-4e00-bbe4-b253e09962f8-32_2647_1835_118_116}