Find the equation of the normal to the parabola \(y^2=4ax\) at the point \((at^2,2at)\).

We need to find the gradient of the normal at this point. How could we find the gradient of the curve itself?

Prove that, if \(p^2>8\), two chords can be drawn through the point \((ap^2,2ap)\) which are normal to the parabola at their second points of intersection, and that the line joining these points of intersection meets the axis of the parabola in a fixed point, independent of \(p\).

Can we find values of \(t\) such that \((ap^2,2ap)\) lies on the normal at \(T\)? Can we get a cubic equation for \(t\) in terms of \(p\)? Do we know any of its roots?

You might find the following applet helpful in visualising the normals to the curve.

As we vary \(p\) using the slider, the point \(P = (ap^2, 2ap)\) moves around the curve. All possible normals through \(P\) are shown, in green and purple.