The point \(S\) is a focus of the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\), and \(P\) is a point on the ellipse such that \(PS\) is perpendicular to the axis of \(x\). The tangent and normal to the ellipse at \(P\) meet the axis of \(y\) in \(Q\) and \(R\) respectively. If \(H\) is the other focus of the ellipse prove that \(QR=HP\).

The lengths \(QR\) and \(HP\) are measured on the diagram.

As we vary \(a\) and \(b\), how do the two lengths compare? Can we now turn this into a proof?

What are the foci of an ellipse? What are their coordinates on the \(x\)-axis (if the \(x\)-axis contains the semi major axis)?

Assume that the focus \(S\) is on the positive \(x\)-axis, without any loss of generality. Once we know the coordinates of \(S\), how can we find the coordinates of \(P\)? (Again, let this point be in the first quadrant.)

What is the gradient of the ellipse at an arbitrary point of the ellipse?

We now have \(P\) and a general form for the gradient. What are the equations of the normal and tangent to the curve at this point? What then are the two intercepts \(Q\) and \(R\)?