Chapter 8.5, Problem 36E

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 $e=5$ , $a=\frac{1}{2}$ , $\sqrt{6}$ and $c=\frac{5}{2}$ .

Given information:

The given equation is $r=\frac{12}{1-5\sin \theta }$ .

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 $\left[-7.5,7.5\right]$ by $\left[-8,2\right]$ to plot the graph as follows.

  

Figure (1)

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

  $e=5$

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

  $\begin{array}{c}2a=3-2\\ 2a=1\\ a=\frac{1}{2}\end{array}$

Substitute $5$ for $e$ and $\frac{1}{2}$ for $a$ in equation (1) to find $c$ .

  $\begin{array}{c}c=5\left(\frac{1}{2}\right)\\ =\frac{5}{2}\end{array}$

Substitute $\frac{1}{2}$ for $a$ and $\frac{5}{2}$ for $c$ in equation (2) to find $b$ .

  $\begin{array}{c}b=\sqrt{{\left(\frac{5}{2}\right)}^2-{\left(\frac{1}{2}\right)}^2}\\ =\sqrt{\frac{25}{4}-\frac{1}{4}}\\ =\sqrt{\frac{24}{4}}\\ =\sqrt{6}\end{array}$

Therefore, the graph is shown in figure (1), and the values are $e=5$ , $a=\frac{1}{2}$ , $\sqrt{6}$ and $c=\frac{5}{2}$ .