Answer
The conic section is an ellipse and the directrix is $x=8/3$.
Work Step by Step
Converting the given equation to the standard form
$$ r=\frac{e d}{1+e \cos \theta}. $$
We get $$ r=\frac{8}{4+ 3\cos \theta}=\frac{2}{1+(3/4) \cos \theta}. $$
Thus $ e=\frac{3}{4}.$ Since $e\lt 1$ then the conic section is an ellipse. To find the directrix, we find $d$ first. Since $ed=2$ and $e=3/4$ then $d=8/3$ and hence the directrix is $x=8/3$.
