Problem

8. The point $P(2p^2, 4p)$ lies on the parabola with equation $y^2 = 8x$

(a) Show that the point $Q\left(\dfrac{2}{p^2}, -\dfrac{4}{p}\right)$, where $p \neq 0$, lies on the parabola.

(b) Show that the chord $PQ$ passes through the focus of the parabola.

The tangent to the parabola at $P$ and the tangent to the parabola at $Q$ meet at the point $R$

(c) Determine, in simplest form, the coordinates of $R$