Review question

# Can we show the locus of the midpoint of $PQ$ is an ellipse? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8051

## Question

The points $P$ and $Q$ on the ellipse

$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$

have coordinates $(a\cos\theta, b\sin \theta)$ and $(-a\sin\theta, b\cos\theta)$ respectively. Show that, if $O$ is the origin,

1. $OP^2+OQ^2 = a^2 + b^2$,
2. the area of triangle $OPQ$ is $\dfrac{1}{2}ab$,
3. the midpoint of $PQ$ always lies on the curve whose equation is

$\dfrac{2x^2}{a^2}+\dfrac{2y^2}{b^2}=1.$