Question

$e=5$; directrix is perpendicular to the polar axis $5$units to

the right of the pole.

Short answer

The polar equation is

$r=\frac{25}{1+5\cos \left(\theta \right)}$

Step-by-step solution

Step 1. Given information 

$e=5$;

The directrix is perpendicular to the polar axis $5$units to

the right of the pole.

Step 2. Identify e and p

It is given that

$e=5$

The directrix is perpendicular to the polar axis 5 units to

the right of the pole.

So,$p=5$

Step 3. Set up the polar equation

It is given that focus is at the pole.

The directrix is perpendicular to the polar axis 5 units to the right of the pole.

So, we can apply the standard polar equation formula

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

Plug values $e=5,p=5$

So, the polar equation is

$$\begin{gathered} r=\frac{5\times 5}{1+5\cos \left(\theta \right)} \\ r=\frac{25}{1+5\cos \left(\theta \right)} \end{gathered}$$