Chapter 11.2, Problem 88E

To determine

To evaluate the given equation and using a graphing utility confirm the result.

Answer to Problem 88E

The conic is an elipse.

Explanation of Solution

Given:

  $r=\frac{12}{3+2\sin \theta }$

calculation:

The given equation should be written in the standard form. For standard form denominator should be 1.

In the first step the numerator and denominator should be multiply by $\frac{\text{1}}{\text{3}}$

i.e

  $\begin{array}{l}\text{r=}\frac{\text{12}}{\text{3+2sinθ}}\\ =\frac{\text{12}\left(\frac{\text{1}}{\text{3}}\right)}{\text{3}\left(\frac{\text{1}}{\text{3}}\right)\text{+2}\left(\frac{\text{1}}{\text{3}}\right)\text{sinθ}}\\ =\frac{\text{6}}{\text{1+}\left(\frac{\text{2}}{\text{3}}\right)\text{sinθ}}\end{array}$

Because $\text{sinθ}$ is in the denominator and the directrix is Y = P

On comparing to standard form $\text{ e=}\frac{\text{2}}{\text{3}}$

Therefore , the numerator is

  $\begin{array}{l}\text{2=ep}\\ \text{2=}\frac{\text{2}}{\text{3}}\text{p}\\ \text{(}\frac{\text{3}}{\text{2}}\text{)2=(}\frac{\text{3}}{\text{2}}\text{)}\frac{\text{2}}{\text{3}}\text{p}\\ \text{3=p}\end{array}$

Since e <1 , the conic is a Elipse.

The eccentricity is e=1 and the directrix is y=3