Ellipses are of fundamental importance in astronomy because keplerian orbits are elliptical. The properties of an ellipse are determined by two quantities, the length of the semimajor axis $a$ and the eccentricity $\varepsilon$, which is defined by requiring that the distance from the center to either of the two foci of the ellipse be $a \varepsilon$. In terms of the distances $r_1$ and $r_2$ from the respective foci to a point on the ellipse, the equation of the ellipse is $r_1+r_2=2 a$. Use this to show that
(a) The length of the semiminor axis $b$ is related to $a$ and $\varepsilon$ through
$$
b^2=a^2\left(1-\varepsilon^2\right)
$$
(b) In terms of a polar coordinate system centered on one of the foci, the distance from that focus to any point on the ellipse is given by
$$
r(\theta)=\frac{a\left(1-\varepsilon^2\right)}{1+\varepsilon \cos \theta},
$$
where $\theta$ is the polar angle between the semimajor axis and the line joining the focus to the point on the ellipse, and where $0 \leq$ $\varepsilon<1$.
(c) The area of an ellipse is $A=\pi a b$. Use the preceding results and Kepler's second law for planetary motion to show that in a time interval $d t$ the planet travels an angular distance measured from the focus