The points \(P\) and \(Q\) on the ellipse


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


Could we start by sketching the graph and marking the points \(P\) and \(Q\)?

When finding the area of the triangle \(OPQ\), are there any other areas we could find more easily that could help us?