Chapter 10.5, Problem 43PE

(a)

To determine

To prove: $c=ea$ , where $e$ is the eccentricity and $a$ represents the length of the semimajor axis of an ellipse.

(b)

To determine

To prove: $r_1=\sqrt{4e^2{a}^2-4ear\cos \theta +r^2}$ , where $e$ is the eccentricity, $a$ represents the length of the semimajor axis, and $r\text{ and }{r}_1$ represent the distances between the foci and the point $P$ on the ellipse.

(c)

To determine

To prove: The ellipse is represented in the polar form by $r=\frac{a\left(1-e^2\right)}{1-e\cos \theta }$ using the definition of an ellipse that states $r+r_1=2a$ .