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