# Thread: Equation of locus of two tangents

1. ## Equation of locus of two tangents

If the normal at $P(ap^2,2ap)$ to the parabola $y^2=4ax$ meets the curve again at $Q(aq^2,2aq)$ prove that $p^2+pq+2=0$. Prove that the equation of the locus of the point of intersection of he tangents to the parabola at P and Q is
$y^2(x+2a)+4a^3=0$.

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

2. Originally Posted by arze
If the normal at $P(ap^2,2ap)$ to the parabola $y^2=4ax$ meets the curve again at $Q(aq^2,2aq)$ prove that $p^2+pq+2=0$. Prove that the equation of the locus of the point of intersection of he tangents to the parabola at P and Q is
$y^2(x+2a)+4a^3=0$.

I've done the first part of proving $p^2+pq+2=0$.
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!
You should find that the point of intersection is P = (apq,a(p+q)). But you know that $p^2+pq+2=0$. Solve that for q, and substitute that value for q into the coordinates of P. That gives $P = (-ap^2-2a,-2a/p)$. So if P = (x,y) then $x = -ap^2-2a$ and $y = -2a/p$. Now eliminate p from those equations for x and y.

3. Originally Posted by Opalg
You should find that the point of intersection is P = (apq,a(p+q)). But you know that $p^2+pq+2=0$. Solve that for q, and substitute that value for q into the coordinates of P. That gives $P = (-ap^2-2a,-2a/p)$. So if P = (x,y) then $x = -ap^2-2a$ and $y = -2a/p$. Now eliminate p from those equations for x and y.
Thanks! But how to eliminate p? Can you explain abit more? thanks

4. Originally Posted by arze
Thanks! But how to eliminate p? Can you explain abit more? thanks
Make p the subject in the parametric equation for y and substitute the resulting expression into the parametric equation for x.