Answer
$r= \dfrac{6}{1 + \sin (θ)}$
Work Step by Step
The polar equation of a conic with eccentricity $e$ and a vertical line directrix is:
$r= \dfrac{ed}{1 + e \sin (θ)}$
Given; $e = 1$ since it is a parabola, Vertex:$(3, \dfrac{\pi}{2} )$
Plug $(3, \dfrac{\pi}{2} )$.
we have $r= \dfrac{ed}{1 + e \sin (θ)} \implies 3= \dfrac{d}{1 + \ sin (\pi /2) }$
$3= \dfrac{d}{2}$
$\implies d=6$
Hence, $r= \dfrac{6}{1 + \sin (θ)}$
