Chapter 5.4, Problem 49E

(a).

To determine

To show: The average value of $d^2=\left(\frac{2b^2+a^2}{3}\right)$ where, $d^2$ is the square of the distance from a planet to the center of the ellipse (0, 0).

(b).

To determine

To show: The average value of the $d^2$ is $R^2$ in part (a), if $a=b=R$.

(c).

To determine

To show: The average value of $D^2=\left(\frac{4a^2-b^2}{3}\right)$, where $D^2$ is the square of the distance from the planet to the Sun.