How do I show that a parametric equation intersects the directrix?

Question

The question was:

The points P and Q on the curve: $$x = 2at, y= at^2$$ have parameters p and q respectively. Show that PQ intersects the directrix at: $$ \left (\frac{2a(pq-1)}{p+q},-a \right ) $$

I've managed to find that the equation of the chord PQ is: $$ y - \frac{1}{2} (p+q)x+apq=0 $$ but after this I'm a bit confused has to how to find the directrix using a parametric equation.

Answer 1

You're there. To proceed, here's a hint.

Hint

This is a standard parabola given by $x^2=4ay$ and its directrix is at $y=-a$. So,

to solve for the point of intersection $PQ$ with $y=-a$ is to just plug in $y=-a$ and solve the resulting linear equation in $x$ (for $x$, of course.)