Answer
$r=\dfrac{2}{4-\cos \theta}$
Work Step by Step
The equation of conic with eccentricity $e$ and directrix $x=k$ leads to focus can be written as:
$r=\dfrac{ke}{1+e \cos \theta}$ ....(1)
Given: $e=\dfrac{1}{4},k=2$
Then $x=-2$
Thus, the equation (1) can be written as:
$r=\dfrac{ke}{1+e \cos \theta}=\dfrac{(\dfrac{1}{2})}{1-(\dfrac{1}{4})\cos \theta}$
or, $r=\dfrac{2}{4-\cos \theta}$