Question

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

$r=\frac{3}{1-\sin \left(\theta \right)}$

Short answer

This polar equation is a parabola.

The equation of directrix is$y=-3$

Step-by-step solution

Step 1. Given information

The polar equation is

$r=\frac{3}{1-\sin \left(\theta \right)}$

Step 2. Finding e and d

The polar equation is

$r=\frac{3}{1-\sin \left(\theta \right)}$

Compare with the standard polar equation

role="math" localid="1647085546390" $r=\frac{ed}{1-e\sin \left(\theta \right)}$

So,

$$\begin{gathered} ed=3 \\ e=1 \end{gathered}$$

Find $d$

$$\begin{gathered} 1\times d=3 \\ d=3 \end{gathered}$$

Step 3. Identify the conic equation

We have got

$e=1$

So, this is a parabola equation.

Step 4. Finding the directrix equation

The formula for the directrix equation is

$y=-d$

Plug $d=3$

So, the directrix equation is$y=-3$.