I'm trying to come up with the polar equation for an off-centre circle that has a certain radius. I've found a lecture handout (pdf) in which the equation is derived for a circle that is off-centre by exactly its radius.
I tried a similar strategy by going from Cartesian to polar coordinates like this:
$$ \begin{align*} &r^2 = (x - x_0)^2 + y^2 \\ \iff& r^2 = x^2 - 2 \cdot x_0 \cdot x + x_0^2 + y^2 \quad \wedge x^2 + y^2 = r^2 \\ \iff& 0 = - 2 \cdot x_0 \cdot x + x_0^2 \quad \wedge x = r \cdot \cos(\theta) \\ \iff& x_0^2 = 2 \cdot x_0 \cdot r \cdot \cos(\theta) \\ \iff& r = \frac{x_0}{2 \cdot \cos(\theta)} \end{align*} $$
But when I try to plot this in Desmos, it yields a vertical line through $x=x_0$.
Am I doing something wrong? Is this even possible?