Answer
$r=\dfrac{4}{3+\cos \theta}$
Work Step by Step
The standard polar equation is: $r=\dfrac{ed}{1+e \cos \theta}$ when the directrix $x=d$
Given: $e=\dfrac{1}{3}$ and the directix is: $r=4 \sec \theta$
Re-arrange as: $r=\dfrac{4}{\cos \theta} \implies r \cos \theta =4$
This gives: $x=4$ or, $x=d=4$
Plug the values in the standard polar equation $r=\dfrac{ed}{1+e \cos \theta}$
we get
$r=\dfrac{(1/3)(4)}{1+(1/3) \cos \theta}$
This implies that
$r=\dfrac{4}{3+\cos \theta}$
