Chapter 8.5, Problem 31E

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 $e=\frac{2}{5}$ , $a=5$ , $b=\sqrt{21}$ and $c=2$ .

Given information:

The given equation is $r=\frac{21}{5-2\cos \theta }$ .

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 $\left[-9,9\right]$ by $\left[-6,6\right]$ to plot the graph as follows.

  

Figure (1)

Divide the numerator and denominator of the given equation by $5$ .

  $r=\frac{\frac{21}{5}}{1-\frac{2}{5}\cos \theta }$

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

  $e=\frac{2}{5}$

The endpoints of the major axis are $\left(7,0\right)$ and $\left(3,\pi \right)$ so the major axis length is as follows.

  $\begin{array}{c}2a=7+3\\ 2a=10\\ a=5\end{array}$

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

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

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

  $\begin{array}{c}b=\sqrt{5^2-2^2}\\ =\sqrt{25-4}\\ =\sqrt{21}\end{array}$

Therefore, the graph is shown in figure (1), and the values are $e=\frac{2}{5}$ , $a=5$ , $b=\sqrt{21}$ and $c=2$ .