The normal at a point \(P\) on the parabola \(y^2=4ax\) meets the ellipse \(2x^2+y^2=c^2\) in the points \(M\), \(N\). Prove that \(P\) is the mid-point of \(MN\).

In this applet you can change the parameters of the parabola and the ellipse. The green line is the normal to the parabola at the point \(P = (at^2, 2at)\).

How could we find the gradient of the parabola and the equation of its normal?

We only need to show that \(P\) is the midpoint of \(MN\) – could we do this without finding M and N?

Hence, or otherwise, prove that the conics cut at right angles.

What happens when we move the points \(M\) and \(N\) near to the point of intersection?