Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.5 Conic Sections - Exercises - Page 636: 51

$$ r= \frac{3}{2+ \cos \theta}. $$

The polar equation of a conic of eccentricity $e \gt0$, with focus at the origin and directrix $x = d$ is $$ r=\frac{e d}{1+e \cos \theta}. $$ Now, since $e=1/2$ and $x=3$, then the polar equation is $$ r=\frac{3/2}{1+(1/2) \cos \theta}=\frac{3}{2+ \cos \theta}. $$