Consider the parabola with equation $y^2 = 4 x$, parameterized by $(x,y) = (t^2,2t)$. If the normals at points corresponding to distinct parameter values $t_1$ and $t_2$ meet at a point on the parabola, find the relation between $t_1$ and $t_2$.
(Note from Blue: It's possible, even likely, that the question intends the intersection to be a distinct third point on the parabola.)
Let the points be $(t_1^2,2t_1)$ and $(t_2^2,2t_2)$.
By taking the derivative at $t_1$ and $t_2$, I get the slopes of the normals as $-t_1$ and $-t_2$.
I then used point-slope form to get the equation of the two normals, and found the $x-$coordinate of intersection as $$x=t_1^2+t_2^2+t_1t_2+2$$
I am stuck over here.