Question

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

$r=\frac{6}{8+2\sin \left(\theta \right)}$

Short answer

This polar equation is a hyperbola.

The equation of directrix is$y=3$

Step-by-step solution

Step 1. Given information

The polar equation is

$r=\frac{6}{8+2\sin \left(\theta \right)}$

Step 2. Finding e and p

The polar equation is

$r=\frac{6}{8+2\sin \left(\theta \right)}$

Divide top and bottom by $8$

$$\begin{gathered} r=\frac{{\displaystyle \frac{6}{8}}}{{\displaystyle \frac{8}{8}}+{\displaystyle \frac{2}{8}}s\mathrm{in}\left(\theta \right)} \\ r=\frac{{\displaystyle \frac{3}{8}}}{{\displaystyle 1+\frac{1}{4}s\mathrm{in}\left(\theta \right)}} \end{gathered}$$

Compare with the standard polar equation

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

So,

role="math" localid="1647172232733" $$\begin{gathered} ep=\frac{3}{4} \\ e=\frac{1}{4} \\ \frac{1}{4}\times p=\frac{3}{4} \\ p=3 \end{gathered}$$

Step 3. Identify the conic equation

We have got

$e=\frac{1}{4}<1$

So, this is an ellipse equation.

Step 4. Finding the directrix equation

The formula for the directrix equation is

$y=p$

Plug value $p=3$

So, the directrix equation is$y=3$