Chapter 10.6, Problem 19AYU

In Problems 13-24, analyze each equation and graph it.

$r=\frac{8}{2-\sin \theta }$

To determine

To graph: Analyze the equation and graph $r=\frac{8}{2-\text{ sin}\theta }$.

Answer to Problem 19AYU

Explanation of Solution

Given:

$r=\frac{8}{2-\text{ sin}\theta }$

Graph:

The graph of $r=\frac{8}{2-\text{ sin}\theta }$,

$r=\frac{8}{2-\text{ sin}\theta }$ can be rewritten as $r=\frac{4}{1-\frac{1}{2}\text{sin}\theta }$.

The above equation is of the form $r=\frac{ep}{1-e\text{sin}\theta }$.

$ep=4$, $e=\frac{1}{2},p=8$

Since $e=\frac{1}{2}$ and $p=8$, the curve represents an ellipse and the directrix is parallel to the polar axis 8 units below the pole. Vertex is represented by $\left(8,\frac{\pi }{2}\right)$ and $\left(\frac{8}{3},\frac{3\pi }{2}\right)$.

The directrix, vertex is shown in the graph.