Answer
Hyperbola
The directrix is parallel to the polar axis at a distance of $\dfrac{4}{3}$ units below the pole.
Work Step by Step
We are given the equation in polar coordinates:
$r=\dfrac{4}{2-3\sin\theta}$
Rewrite the equation:
$r=\dfrac{\dfrac{4}{2}}{\dfrac{2-3\sin\theta}{2}}$
$r=\dfrac{2}{1-\dfrac{3}{2}\sin\theta}$
The equation is in the form:
$r=\dfrac{ep}{1-e\sin\theta}$
Identify $e$ from the denominator, then $p$ from the numerator:
$e=\dfrac{3}{2}$
$ep=2\Rightarrow \dfrac{3}{2}p=2\Rightarrow p=\dfrac{4}{3}$
Because $e>1$, the conic is a hyperbola. The directrix is parallel to the polar axis at a distance of $\dfrac{4}{3}$ units below the pole.