Properties of Ellipses, Parabolas & Hyperbolas (Edexcel A Level Further Maths: Further Pure 1): Exam Questions

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

The point $A(a{t}^2,2at)$, where $t\ne 0$, lies on $P$.

Use calculus to show that an equation of the normal to $P$ at $A$ is

$y+tx=2at+a{t}^3$

The point $B(3k^2,6k)$ and the point $C(3k^2,-6k)$, where $k$ is a constant, lie on $P$.

The normal to $P$ at $B$ and the normal to $P$ at $C$ intersect at the point $D$.

Given that the area of triangle $BCD$ is 108, determine the coordinates of $B$ and the coordinates of $C$.