Chapter 11.6, Problem 16E

To determine

To draw: the graph of the polar equations.

Explanation of Solution

Given information:

The polar equation is $r=\frac{12}{2+3\cos \theta }$ .

Graph: first write the equation in the standard equation,

  $r=\frac{\frac{3}{2}\left(8\right)}{1+\frac{3}{2}\cos \theta }$

since, the polar equation of conic is $r=\frac{\frac{3}{2}\left(8\right)}{1+\frac{3}{2}\cos \theta }$ , so the $e=\frac{3}{2}\text{ and }d=8$ ,

Thus, the conic is hyperbola which directrix is $x=8$ .

Now, by using plotting point method,

$\theta$	$r$
$0$	$\frac{12}{5}$
$\frac{\pi }{2}$	$6$
$\frac{\pi }{3}$	$\frac{24}{7}$

Now, using the above table and join the points,

The graph can be obtained as:

  

Interpretation: from the above graph it can be observed that for the polar equation the conic is hyperbola.