Chapter 8.5, Problem 33E

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 $e=\frac{1}{2}$ , $a=8$ , $b=4\sqrt{3}$ and $c=4$ .

Given information:

The given equation is $r=\frac{24}{4+2\sin \theta }$ .

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 $\left[-15,5\right]$ to plot the graph as follows.

  

Figure (1)

Divide the numerator and denominator of the given equation by $4$ .

  $r=\frac{6}{1+\frac{1}{2}\sin \theta }$

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

  $e=\frac{1}{2}$

The endpoints of the major axis are $\left(4,\frac{\pi }{2}\right)$ and $\left(12,\frac{3\pi }{2}\right)$ so the major axis length is as follows.

  $\begin{array}{c}2a=4+12\\ 2a=16\\ a=8\end{array}$

Substitute $\frac{1}{2}$ for $e$ and $8$ for $a$ in equation (1) to find $c$ .

  $\begin{array}{c}c=\frac{1}{2}\left(8\right)\\ =4\end{array}$

Substitute $8$ for $a$ and $4$ for $c$ in equation (2) to find $b$ .

  $\begin{array}{c}b=\sqrt{8^2-4^2}\\ =\sqrt{64-16}\\ =\sqrt{48}\\ =4\sqrt{3}\end{array}$

Therefore, the graph is shown in figure (1), and the values are $e=\frac{1}{2}$ , $a=8$ , $b=4\sqrt{3}$ and $c=4$ .