Review question

# When do these normals to a parabola meet on the parabola? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9809

## Suggestion

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

How can we find the gradient of the parabola at a point with parameter $t$?

If we have the gradient of the normal through the point $(at^2,2at)$, what is the equation of the line?

The parameters of the points $P$, $Q$ are $t_1$ and $t_2$ respectively. Show that, if $PQ$ passes through the point $(-2a,0)$, then $t_1t_2=2$

What is the gradient of the line that passes through $P$ and $Q$? What about its equation?

…and the normals at $P$ and $Q$ to the parabola meet at a point $R$ on the parabola.

Could we write down the equations of the normals at $P$ and $Q$, and find where they intersect?

If $O$ is the origin, show, by considering the gradients of the sides of the quadrilateral $OPQR$ or otherwise, that the circumcircle of the triangle $PQR$ passes through $O$.

Aren’t we really being asked to show that $OPQR$ is a cyclic quadrilateral? What do we know about cyclic quadrilaterals?