I didn't like your parameterisation, since for the circle
x2+y2−2x=0 there are two y values for a given x value, which you don't get.
I used
y=sint,x=1−cost where t is the angle from (1,0) to the centre of the parameterised circle.
This gave me a
tant=−xy from the partial derivative.
My final equation was
r=2(cosθ−2), which although different to the given answer, does yield the same curve.
The working was rather horrendous, and I don't know why I've ended up with a different representation of the curve.
Edit: Correct point to (1,0)