Chapter 8.5, Problem 22E

Solution

To find: The polar equation for the ellipse with a focus at the pole and the given polar coordinates as the endpoints of its major axis.

The required equation is $r=\frac{6}{5-\cos \theta }$ .

Given information:

The points are $\left(1.5,0\right)$ and $\left(1,\pi \right)$ .

Formula used:

The focus at the pole is nearer to the point $\left(1,\pi \right)$ and the directrix is vertical, to the left of the pole, so use the following polar equation.

  $r=\frac{ke}{1-e\cos \theta }$   ..... (1)

Calculation:

For illustration purposes, the locations of the endpoints of the major axis are on the polar axis as shown below:

  

Substitute $1.5$ for $r$ and $0$ for $\theta$ in equation (1).

  $\begin{array}{c}1.5=\frac{ke}{1-e\cos 0}\\ 1.5=\frac{ke}{1-e}\\ 1.5-1.5e=ke\end{array}$   ..... (2)

Substitute $1$ for $r$ and $\pi$ for $\theta$ in equation (1).

  $\begin{array}{c}1=\frac{ke}{1+e\cos \pi }\\ 1=\frac{ke}{1+e}\\ 1+e=ke\end{array}$   ..... (3)

Subtract equation (2) and (3).

  $\begin{array}{c}\left(1.5-1.5e\right)-\left(1+e\right)=ke-ke\\ 0.5-2.5e=0\\ 2.5e=0.5\\ e=\frac{1}{5}\end{array}$

Substitute $\frac{1}{5}$ for $e$ in equation (3) to find $k$ .

  $\begin{array}{c}1+\left(\frac{1}{5}\right)=k\left(\frac{1}{5}\right)\\ k=\frac{1+0.2}{0.2}\\ =6\end{array}$

Substitute $\frac{1}{5}$ for $e$ and $6$ for $k$ in equation (1).

  $\begin{array}{c}r=\frac{\left(6\right)\left(\frac{1}{5}\right)}{1-\frac{1}{5}\cos \theta }\\ r=\frac{6}{5-\cos \theta }\end{array}$

Therefore, the required equation is $r=\frac{6}{5-\cos \theta }$ .