Question

$e=\frac{2}{3}$; directrix is parallel to the polar axis $3$units above the

pole.

Short answer

The polar equation is

$r=\frac{6}{3+2\sin \left(\theta \right)}$

Step-by-step solution

Step 1. Given information 

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

The directrix is parallel to the polar axis $3$units above the

pole.

Step 2. Identify e and p

It is given that

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

And directrix is parallel to the polar axis $3$units above the

pole.

So,$p=3$

Step 3. Set up the polar equation

It is given that focus is at the pole.

The directrix is parallel to the polar axis $3$units above the pole.

So, we can apply standard polar equation formula

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

Plug values $e=\frac{2}{3},p=3$

$$\begin{gathered} r=\frac{{\displaystyle \frac{2}{3}}\times 3}{1+\frac{2}{3}\sin \left(\theta \right)} \\ r=\frac{{\displaystyle 2\times 3}}{3(1+\frac{2}{3}\sin (\theta \left)\right)} \\ r=\frac{{\displaystyle 6}}{3+2\sin \left(\theta \right)} \end{gathered}$$