P is any point on an ellipse of which the foci are S and H. The distance SP is denoted by \(r\) and the angle HSP by \(\theta\). Show that the mean value of \(r\) with respect to arc is the semi-major axis \(a\), and that the mean value of \(r\) with respect to \(\theta\) is the semi-minor axis \(b\). If Q is any point in the interior of the ellipse, show that the mean value of the distance SQ with respect to area is \(a - \frac{b^2}{3a}\).