Chapter 10.6, Problem 24AYU

To determine

To graph: Analyze the equation and graph $r=\frac{3\text{csc}\theta }{\text{csc}\theta -1}$.

Answer to Problem 24AYU

Explanation of Solution

Given:

$r=\frac{3\text{csc}\theta }{\text{csc}\theta -1}$

Graph:

The graph of $r=\frac{3\text{csc}\theta }{\text{csc}\theta -1}$.

$r=\frac{3\text{csc}\theta }{\text{csc}\theta -1}$ can be rewritten as $r=\frac{3}{1-\text{ sin}\theta }$.

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

$ep=3$, $e=1,p=3$

Since $e=\frac{1}{2}$ and $p=6$, the curve represents a parabola and the directrix is parallel to the polar axis 3 units below the pole. Vertex is represented by $\left(\frac{3}{2},\frac{3\pi }{2}\right)$.

The directrix, vertices is shown in the graph.