Question

True or False:If $(r,\theta )$are polar coordinates, the equation $r=\frac{2}{2+3\sin \left(\theta \right)}$defines a hyperbola.

Short answer

This statement is True.

Step-by-step solution

Step 1. Given information

The polar equation is

$r=\frac{2}{2+3\sin \left(\theta \right)}$

Step 2. Checking the statement

The polar equation is

$r=\frac{2}{2+3\sin \left(\theta \right)}$

Divide top and bottom by $2$

$$\begin{gathered} r=\frac{{\displaystyle \frac{2}{2}}}{{\displaystyle \frac{2}{2}}+{\displaystyle \frac{3}{2}}\sin \left(\theta \right)} \\ r=\frac{1}{1+{\displaystyle \frac{3}{2}}\sin \left(\theta \right)} \end{gathered}$$

Compare with the standard polar equation

$r=\frac{ep}{1+e\sin \left(\theta \right)}$

So,

$e=\frac{3}{2}>1$

Therefore, this polar equation is a hyperbola.