Question 6 - A-Level Maths

The parabola $C$ has equation $y ^ { 2 } = 32 x$ and the point $S$ is the focus of this parabola. The point $P ( 2,8 )$ lies on $C$ and the point $T$ lies on the directrix of $C$. The line segment $P T$ is parallel to the $x$-axis.
1. Write down the coordinates of $S$.
2. Find the length of $P T$.
3. Using calculus, show that the tangent to $C$ at the point $P$ has equation
$$y = 2 x + 4$$ The hyperbola $H$ has equation $x y = 4$. The tangent to $C$ at $P$ meets $H$ at the points $L$ and $M$.
Find the exact coordinates of the points $L$ and $M$, giving your answers in their simplest form.