In Complex Analysis, if I want to prove that a certain function maps any circle through point (-1,0) into a line, is there a convenient way to express all these circles?

The function in question is f(z)=(z-1)/(iz+i). My book goes through an example where they show that this function maps the unit circle (which can of course be expressed as e^(i*phi) ) to a line.

I wish to prove that any circle through the point (-1,0) will also be mapped to a line. Is there a general, convenient way to express this set in polar coordinates, such that I can use it as input for the function f(z) and show that the output is a line? Or do I need to explore a different approach?

Thanks very much for your input!