Review question

# What can we say if a point $P$ on an ellipse is directly above the focus? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6710

## Suggestion

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 $R$?