Can we show the locus of the midpoint of $PQ$ is an ellipse?

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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

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

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?