I'm having trouble describing the set: $\{z\in\mathbb{C}:|z-a|=r|z-b|\}$ where $r$ is a positive real number and $a,b$ are fixed complex numbers. I worked out the algebra and it seems to be a (real) equation in two variables each with maximum degree $2$. This seems to imply that it is some sort of conic section depending on the values of $a,b,r$. Is this correct?
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If $a \neq b$ and $r \neq 1$ then your set is a circle.
If $a \neq b$ and $r=1$ then your set is a line: the perpendicular bisector of $a$ and $b$.
If $a = b$ and $r \neq 1$ then your set is the single point $\{a\}=\{b\}$
If $a=b$ and $r = 1$ then your set is all of the complex plane.
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