Question

$e=\frac{4}{5}$; directrix is perpendicular to the polar axis $3$units to

the left of the pole.

Short answer

The polar equation is

$r=\frac{12}{5-4\cos \left(\theta \right)}$

Step-by-step solution

Step 1. Given information 

$e=\frac{4}{5}$

The directrix is perpendicular to the polar axis $3$units to

the left of the pole.

Step 2. Identify e and p

It is given that

$e=\frac{4}{5}$

directrix is perpendicular to the polar axis $3$units to

the left of the pole.

So,$p=3$

Step 3. Set up the polar equation

It is given that focus is at the pole.

directrix is perpendicular to the polar axis $3$units to the left of the pole.

So, we can apply standard polar equation formula

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

Plug values $e=\frac{4}{5},p=3$

$$\begin{gathered} r=\frac{{\displaystyle \frac{4}{5}}\times 3}{1-{\displaystyle \frac{4}{5}}\cos \left(\theta \right)} \\ r=\frac{{\displaystyle 4\times 3}}{5(1-{\displaystyle \frac{4}{5}}\cos (\theta \left)\right)} \\ r=\frac{{\displaystyle 12}}{5-{\displaystyle 4}\cos \left(\theta \right)} \end{gathered}$$