Question

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

$r=\frac{2}{1+2\cos \left(\theta \right)}$

Short answer

This polar equation is a hyperbola.

The equation of directrix is$x=1$

Step-by-step solution

Step 1. Given information

The polar equation is

$r=\frac{2}{1+2\cos \left(\theta \right)}$

Step 2. Finding e and p

The polar equation is

$r=\frac{2}{1+2\cos \left(\theta \right)}$

Compare with the standard polar equation

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

So,

$$\begin{gathered} ep=2 \\ e=2 \\ 2\times p=2 \\ p=1 \end{gathered}$$

Step 3. Identify the conic equation

We have got

$e=2>1$

So, this is a hyperbola equation.

Step 4. Finding the directrix equation

The formula for the directrix equation is

$x=p$

Plug values $p=1$

So, the directrix equation is$x=1$