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

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