Answer
III
Work Step by Step
The equation of conic with eccentricity $e$ and directrix $d=x$ leads to focus can be written as:
$r=\dfrac{d}{1 -\cos \theta}$ when $e=1$
As we are given $r=\dfrac{d}{1-0.5\cos \theta}$
When we look at all the graphs it has been noticed when $e=0.5 \lt 1$ . This shows an ellipse, so the correct graph is III.
