Chapter 1.3, Problem 203E

A point on an ellipse with major axis length 2a and minor axis length 2b has the coordinates $\left(a\cos \theta ,b\sin \theta \right),0\le \theta \le 2\pi$ .

a. Show that the distance from this point to the focus at (—c, 0) is $d\left(\theta \right)=a+c\cos \theta$ , where $c=\sqrt{a^2-b^2}$ .

b. Use these coordinates to show that the average distance $\bar{d}$ from a point on the ellipse to the focus at (−c, 0), with respect to angle (, is a.