I have been given a question which is very similar to this:
Apollonius circle, its radius and center
However, I have been told to translate and scale the given circle to give a unit circle centred at the origin, and then to show that for any point $a \in\Bbb{C}$ there is a unique $b \in \Bbb{C}$ and $k \in (0,1)$ such that $|z-a| = k|z-b|$.
Is there any way to do this without completing the square?
My lecturer told us to get it to the unit circle centred at the origin where he took $a = 1/2$, and we need to tackle it like this - but am I allowed to take $a =1/2$ or is this the generic case?
Update: I've managed to get a = 1/b and b = 1/a, but still cannot find k
