Chapter 8.5, Problem 32E

Solution

To plot: The graph of the given conic equation and find the values of $e,a,b$ and $c$ .

The graph is shown in figure (1), and the values are $e=\frac{5}{6}$ , $a=6$ , $b=\sqrt{11}$ and $c=5$ .

Given information:

The given equation is $r=\frac{11}{6-5\sin \theta }$ .

Formula used:

The formula to calculate the value of $c$ is given by,

  $c=ea$   ..... (1)

The formula to calculate the value of $b$ is given by,

  $\begin{array}{c}{a}^2=b^2+c^2\\ b=\sqrt{a^2-c^2}\end{array}$   ..... (2)

Calculation:

Use a graphing calculator in polar mode; enter the given conic, and adjusting the window as $\left[-15,15\right]$ by $\left[-5,15\right]$ to plot the graph as follows.

  

Figure (1)

Divide the numerator and denominator of the given equation by $6$ .

  $r=\frac{\frac{11}{6}}{1-\frac{5}{6}\sin \theta }$

Compare the above equation with $r=\frac{ke}{1-e\sin \theta }$ .

  $e=\frac{5}{6}$

The endpoints of the major axis are $\left(11,\frac{\pi }{2}\right)$ and $\left(1,\frac{3\pi }{2}\right)$ so the major axis length is as follows.

  $\begin{array}{c}2a=11+1\\ 2a=12\\ a=6\end{array}$

Substitute $\frac{5}{6}$ for $e$ and $6$ for $a$ in equation (1) to find $c$ .

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

Substitute $6$ for $a$ and $5$ for $c$ in equation (2) to find $b$ .

  $\begin{array}{c}b=\sqrt{6^2-5^2}\\ =\sqrt{36-25}\\ =\sqrt{11}\end{array}$

Therefore, the graph is shown in figure (1), and the values are $e=\frac{5}{6}$ , $a=6$ , $b=\sqrt{11}$ and $c=5$ .