Problem

8. The parabola $C$ has equation $y^2 = 20x$

The point $P$ on $C$ has coordinates $(5p^2, 10p)$ where $p$ is a non-zero constant.

(a) Use calculus to show that the tangent to $C$ at $P$ has equation

$py - x = 5p^2$

The tangent to $C$ at $P$ meets the $y$-axis at the point $A$.

(b) Write down the coordinates of $A$.

The point $S$ is the focus of $C$.

(c) Write down the coordinates of $S$.

The straight line $l_1$ passes through $A$ and $S$.

The straight line $l_2$ passes through $O$ and $P$, where $O$ is the origin.

Given that $l_1$ and $l_2$ intersect at the point $B$,

(d) show that the coordinates of $B$ satisfy the equation

$2x^2 + y^2 = 10x$