Question

Identify the conic that each polar equation represents. Also, give the position of the directrix.

$r=\frac{4}{2-3\sin \left(\theta \right)}$

Short answer

This polar equation is a hyperbola.

The equation of directrix is$y=-\frac{4}{3}$

Step-by-step solution

Step 1. Given information

The polar equation is

$r=\frac{4}{2-3\sin \left(\theta \right)}$

Step 2. Finding e and p

The polar equation is

$r=\frac{4}{2-3\sin \left(\theta \right)}$

Divide top and bottom by $2$

$$\begin{gathered} r=\frac{{\displaystyle \frac{4}{2}}}{{\displaystyle \frac{2}{2}}-{\displaystyle \frac{3}{2}}\sin \left(\theta \right)} \\ r=\frac{2}{1-{\displaystyle \frac{3}{2}}\sin \left(\theta \right)} \end{gathered}$$

Compare with the standard polar equation

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

So,

$$\begin{gathered} ep=2 \\ e=\frac{3}{2} \\ \frac{3}{2}\times p=2 \\ p=\frac{4}{3} \end{gathered}$$

Step 3. Identify the conic equation

We have got

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

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

So, this is a hyperbola equation.

Step 4. Finding the directrix equation

The formula for the directrix equation is

$y=-p$

Plug $p=\frac{4}{3}$

So, the directrix equation is$y=-\frac{4}{3}$