Problem

The hyperbola $H$ has equation

$$4x^2 - y^2 = 4.$$

(a) Write down the equations of the asymptotes of $H$.

(b) Find the coordinates of the foci of $H$.

The point $P(\sec\theta, 2\tan\theta)$ lies on $H$.

(c) Using calculus, show that the equation of the tangent to $H$ at the point $P$ is

$$y\tan\theta = 2x\sec\theta - 2.$$

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 $VP$ and $WQ$ intersect at the point $S$,

(d) 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.