Chapter 9.7, Problem 16E

To determine

To identify:

Type of conic represented by the equation using graphing utility.

Answer to Problem 16E

Type of conic represented by the equation using graphing utility ellipse.

Explanation of Solution

Given information:

  $r=\frac{2}{4+\sin \theta }$

Consider the following equation in polar form,

  $\begin{array}{l}r=\frac{2}{4+\sin \theta }\\ =\frac{1}{2+\left(\frac{1}{2}\right)\sin \theta }\end{array}$

Here,

  $e=\frac{1}{2}<1$

Thus, the conic is an ellipse.

Also the equation is an ellipse.

  $r=\frac{ep}{1+e\sin \theta }$

Thus the horizontal directrix will be above the pole.

Since the equation is of the form.

  $r=f\left(\sin \theta \right)$

Therefore, the graph is symmetrical about the line.

  $\theta =\pi /2$

Graph of the conic is as below: