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