Answer
parabola, $y^2=x+\frac{1}{4}$
Work Step by Step
Step 1. Rewrite the equation as $r=\frac{\frac{1}{2}}{1-cos\theta}$. Thus we have $e=1, ep=\frac{1}{2}, p=\frac{1}{2}$ and the curve is a parabola with its directrix as $x=-\frac{1}{2}$
Step 2. Rewrite the equation as $2r-2r\ cos\theta = 1$ and use the relation $r\ cos\theta = x$. We have $2r-2x =1$ or $2r=2x+1$
Step 3. Take the square on both sides and use the relation $r^2=x^2+y^2$. We have $4(x^2+y^2)=(2x+1)^2$, which can be simplified to $4y^2=4x+1$ or in standard form $y^2=x+\frac{1}{4}$
