Problem

6. The parabola $C$ has equation $y^2 = 36x$

The point $P(9t^2, 18t)$, where $t \neq 0$, lies on $C$

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

$y + tx = 9t^3 + 18t$

(b) Hence find the equations of the two normals to $C$ which pass through the point $(54, 0)$, giving your answers in the form $y = px + q$ where $p$ and $q$ are constants to be determined.

Given that

• the normals found in part (b) intersect the directrix of $C$ at the points $A$ and $B$
• the point $F$ is the focus of $C$

(c) determine the area of triangle $AFB$