Chapter 10.6, Problem 21AYU

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

$r\left(3-2\sin \theta \right)=6$

To determine

To graph: Analyze the equation and graph $r\left(3-2sin\theta \right)=6$.

Answer to Problem 21AYU

Explanation of Solution

Given:

$r\left(3-2sin\theta \right)=6$

Graph:

The graph of $r\left(3-2sin\theta \right)=6$,

$r\left(3-2sin\theta \right)=6$ can be rewritten as $r=\frac{6}{3-2\text{sin}\theta }=\frac{2}{1-\frac{2}{3}\text{sin}\theta }$.

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

$ep=2$, $e=2/3,p=3$

Since $e=2/3$ and $p=3$, the curve represents an ellipse and the directrix is parallel to the polar axis 3 units below the pole. Vertices are represented by $\left(6,\pi /2\right)$ and $\left(\frac{6}{5},3\pi /2\right)$.

The directrix, vertex is shown in the graph.