So, I basically want to show that points $z$ satisfying $|z - a| = k|z - b|$ (where $a$ and $b$ are also complex) gives me a circle of radius $$\frac{k|a - b|}{|1-k^2|},$$ centred at $$\frac{a-bk^2}{1-k^2}.$$
I tried to boil this down to a cartesian equation that showed me the properties of the circle I was looking for, found the centre by doing this, but still (as you can imagine) ended up in a horrible algebraic mess, with no hope of showing my right hand side was in fact the square of the radius.
Is there a "neater" way of deriving that we have a circle with the above centre and radius?
Many thanks! Sam
Presumably $k > 0$. To reduce the algebraic mess, take advantage of symmetry. You want the centre to be at $a - k^2 b$, so let's translate so that is $0$.
Then rotate and scale so that $b=1$ and $a = k^2$. Now your equation is $|z - k^2| = k |z - 1|$. Squaring both sides, $$ (z - k^2)(\overline{z} - k^2) = k^2 (z - 1)(\overline{z} - 1)$$ Now expand and simplify.