Chapter 8.5, Problem 35E

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 $e=\frac{5}{3}$ , $a=3$ , $b=4$ and $c=5$ .

Given information:

The given equation is $r=\frac{16}{3+5\cos \theta }$ .

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 $\left[-4,14\right]$ by $\left[-6,6\right]$ to plot the graph as follows.

  

Figure (1)

Divide the numerator and denominator of the given equation by $3$ .

  $r=\frac{\frac{16}{3}}{1+\frac{5}{3}\cos \theta }$

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

  $e=\frac{5}{3}$

The endpoints of the major axis are $\left(2,0\right)$ and $\left(8,0\right)$ so the major axis length is as follows.

  $\begin{array}{c}2a=8-2\\ 2a=6\\ a=3\end{array}$

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

  $\begin{array}{c}c=\frac{5}{3}\left(3\right)\\ =5\end{array}$

Substitute $3$ for $a$ and $5$ for $c$ in equation (2) to find $b$ .

  $\begin{array}{c}b=\sqrt{5^2-3^2}\\ =\sqrt{25-9}\\ =\sqrt{16}\\ =4\end{array}$

Therefore, the graph is shown in figure (1), and the values are $e=\frac{5}{3}$ , $a=3$ , $b=4$ and $c=5$ .