Answer
$r=\dfrac{2}{2+ \cos \theta}$
Work Step by Step
The polar equation of a conic with eccentricity $e$ and directrix $x=d$ can be written as:
$r=\dfrac{ed}{1+e \cos \theta}$ ....(1)
The conic will be an ellipse when $e \lt 1$
Given: $e=\dfrac{1}{2}$ and Directrix =4
Thus, the equation (1) can be written as:
$r=\dfrac{ed}{1+e \cos \theta}=\dfrac{(1/2)(4)}{1+(1/2)\cos \theta}=\dfrac{2}{2+ \cos \theta}$
