Answer
hyperbola, $(x+2)^2-\frac{y^2}{3}=1$
Work Step by Step
1. Given $r=\frac{3}{1-2cos\theta}$, we have $e=2\gt1$, thus it is a hyperbola.
2. We have $r-2r\ cos\theta=3\Longrightarrow r^2=(2r\ cos\theta+3)^2 \Longrightarrow x^2+y^2=(2x+3)^2 \Longrightarrow 3x^2+12x-y^2+9=0 \Longrightarrow (x+2)^2-\frac{y^2}{3}=1$
