Chapter 11.6, Problem 15E

To determine

To draw: the graph of the polar equations.

Explanation of Solution

Given information:

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

Graph: first write the equation in the standard equation,

  $r=\frac{\frac{2}{3}\left(6\right)}{1+\frac{2}{3}\sin \theta }$

since, the polar equation of conic is $r=\frac{\frac{2}{3}\left(6\right)}{1+\frac{2}{3}\sin \theta }$ , so the $e=\frac{2}{3}\text{ and }d=6$ ,

Thus, the conic is ellipse which directrix is $y=6$ .

Now, by using plotting point method,

$\theta$	$r$
$0$	$4$
$\frac{\pi }{2}$	$\frac{12}{5}$
$\frac{\pi }{6}$	$3$

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 ellipse.