Chapter 10.6, Problem 20AYU

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

$r=\frac{8}{2+4\cos \theta }$

To determine

To graph: Analyze the equation and graph $r=\frac{8}{2+4\text{cos}\theta }$.

Answer to Problem 20AYU

Explanation of Solution

Given:

$r=\frac{8}{2+4\text{cos}\theta }$

Graph:

The graph of $r=\frac{8}{2+4\text{cos}\theta }$,

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

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

$ep=4$, $e=2,p=2$

Since $e=2$ and $p=2$, the curve represents a hyperbola and the directrix is perpendicular to the polar axis 2 units to the right of the pole. Vertices are represented by $\left(\frac{4}{3},0\right)$ and $\left(-4,\pi \right)$.

The directrix, vertex is shown in the graph.