Question 8 - A-Level Maths

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

Use calculus to show that the tangent to $C$ at $P$ has equation $$p y - x = 5 p ^ { 2 }$$ The tangent to $C$ at $P$ meets the $y$-axis at the point $A$.
Write down the coordinates of $A$. The point $S$ is the focus of $C$.
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 $B$,
show that the coordinates of $B$ satisfy the equation $$2 x ^ { 2 } + y ^ { 2 } = 10 x$$