Chapter 10.6, Problem 16AYU

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

$r=\frac{10}{5+4\cos \theta }$

To determine

To graph: Analyze the equation and graph it.

Answer to Problem 16AYU

Explanation of Solution

Given:

$r=\frac{10}{5+4\text{cos}\theta }$

Graph:

The graph of $r=\frac{10}{5+4\text{cos}\theta }$,

$r=\frac{10}{5+4\text{cos}\theta }$, Rewriting the equation we get,

$r=\frac{\frac{10}{5}}{\frac{5}{5}+\frac{4}{5}\text{sin}\theta }=\frac{2}{1+\frac{4}{5}\text{sin}\theta }$ is of the form $r=\frac{ep}{1+e\text{sin}\theta }$.

$ep=2$, $e=4/5,p=5/2$

Since $e=4/5$ and $p=5/2$, the curve represents an ellipse and the directrix is perpendicular to the polar axis $5/2$ units right of the pole. Vertex is represented by $\left(10,\pi \right)$ and $\left(\frac{10}{9},0\right)$.

The directrix, vertex is shown in the graph.