Answer
$r=\dfrac{8}{1+2\cos \theta}$
Work Step by Step
The polar equation of a conic with eccentricity $e$ and directrix $x=k$ is defined as:
$r=\dfrac{ke}{1+e \cos \theta}$ ...(1)
We are given that the vertices are: $e=2,k=4$
Then $x=4$
Thus, equation (1), becomes
$r=\dfrac{8}{1+2\cos \theta}$
