Question

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

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

Short answer

This polar equation is an ellipse.

The equation of directrix $x=-\frac{3}{2}$

Step-by-step solution

Step 1. Given information

The polar equation is

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

Step 2. Finding e and p

The polar equation is

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

Divide top and bottom by $4$

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

Compare with the standard polar equation

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

So,

$$\begin{gathered} ep=\frac{3}{4} \\ e=\frac{1}{2} \\ \frac{1}{2}\times p=\frac{3}{4} \\ p=\frac{3}{2} \end{gathered}$$

Step 3. Identify the conic equation

We have got

$e=\frac{1}{2}<1$

So, this is an ellipse equation.

Step 4. Finding the directrix equation

The formula for the directrix equation is

$x=-p$

Plug $p=\frac{3}{2}$

So, the directrix equation is$x=-\frac{3}{2}$