Describe the curve $r=\frac{a}{\cos((\theta-b)c)}$ in the $(r,\theta)$ plane

Question

I'm given the curve $$r=\frac{a}{\cos((\theta-b)c)}$$

and I'm asked what this looks like in the $(r,\theta)$ plane. My guess is that it would just be a straight line, but I don't know how to justify this.

How can I justify my answer? (Assuming I'm correct).

Answer 1

It is a straight line if $|c|=1$.

When $c=1$, $\displaystyle \cos(\theta-b)=\frac{a}{r}$. This represents a straight line. If we drop a perpendicular line segment from the pole to the line, this perpendicular line segment has length $a$ and makes an angle $b$ with the polar axis.

When $|c|\ne1$, the curve is not a line.