Review question

# Can we show this parabola and this ellipse meet at right angles? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7042

## Suggestion

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?