Review question

# How many normals to a parabola pass through $P$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5053

## Suggestion

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.