For drawing a parallel line to the x-axis through (0,b), simply use
r(phi) = b/sin(phi)
If you want to have a general linear function ax+b in poloar coordinates, simply note
r^2 = x^2 + (ax+b)^2
x = r cos(phi)
This results in a polynom of second degree in r, which you can already solve for r (as I have heard
)
For verification let a=0, then
r^2 = (r cos(phi))^2 + b^2
r^2(1-cos^2(phi)) = b^2
r^2 sin^2(phi) = b^2
r = b/sin(phi)
and we have the first given formula.
Or did I completely misinterpret your question :?