Chapter 10.6, Problem 23AYU

To determine

To graph: Analyze the equation and graph $r=\frac{6\text{sec}\theta }{2\text{sec}\theta -1}$.

Answer to Problem 23AYU

Explanation of Solution

Given:

$r=\frac{6\text{sec}\theta }{2\text{sec}\theta -1}$

Graph:

The graph of $r=\frac{6\text{sec}\theta }{2\text{sec}\theta -1}$.

$r=\frac{6\text{sec}\theta }{2\text{sec}\theta -1}$ can be rewritten as $r=\frac{6\text{sec}\theta }{2\text{sec}\theta -1}=\frac{3}{1-\frac{1}{2}\text{cos}\theta }$

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

$ep=3$, $e=\frac{1}{2},p=6$

Since $e=\frac{1}{2}$ and $p=6$, the curve represents an ellipse and the directrix is perpendicular to the polar axis 6 units left of the pole. Vertices are represented by $\left(2,\text{ π}\right)\text{ and }\left(6,0\right)$.

The directrix, vertices is shown in the graph.