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?
