Problem

7. The parabola $C$ has equation $y^2 = 4ax$, where $a$ is a positive constant.

The line $l$ with equation $3x - 4y + 48 = 0$ is a tangent to $C$ at the point $P$.

(a) Show that $a = 9$

(b) Hence determine the coordinates of $P$.

Given that the point $S$ is the focus of $C$ and that the line $l$ crosses the directrix of $C$ at the point $A$,

(c) determine the exact area of triangle $PSA$.