Answer
$$
r=\frac{4}{1+ \cos \theta} .
$$
Work Step by Step
The polar equation of a conic of eccentricity $e \gt0$, with focus at the origin and directrix $x = d$ is
$$
r=\frac{e d}{1+e \cos \theta}.
$$
Now, since $e=1$ and $x=4$, then the polar equation is
$$
r=\frac{4}{1+ \cos \theta} .
$$
