Answer
$r=\dfrac{4}{3+\cos \theta}$
Work Step by Step
Given: $e=\dfrac{1}{3}$ and the directrix is: $r=4 \sec \theta$
The standard polar equation for a conic when the directrix $x=d$ is $r=\dfrac{ed}{1+e \cos \theta}$
This can be re-written as:
$r=\dfrac{4}{\cos \theta} \implies r \cos \theta =4$
$\implies x=4$
$\implies x=d=4$
Now, we have
$r=\dfrac{ed}{1+e \cos \theta}=\dfrac{(1/3)(4)}{1+(1/3) \cos \theta}$
Thus,
$r=\dfrac{4}{3+\cos \theta}$
