Equation of locus of two tangents

If the normal at $\displaystyle P(ap^2,2ap)$ to the parabola $\displaystyle y^2=4ax$ meets the curve again at $\displaystyle Q(aq^2,2aq)$ prove that $\displaystyle p^2+pq+2=0$. Prove that the equation of the locus of the point of intersection of he tangents to the parabola at P and Q is

$\displaystyle y^2(x+2a)+4a^3=0$.

I've done the first part of proving $\displaystyle p^2+pq+2=0$.

for this second part, I tried to find a parametric equation for the locus the point of intersection of the tangents. Found the equation of the two tangents, then the point of intersection. The problem I have is there is no p or q in the equation I'm supposed to prove, while the point is in terms of p and q. Need someone to point me in the right direction and I should be able to finish off. Thanks!