Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 7 / ii / Solution

Squaring the ratio of moduli givesFor $\lambda \ne 1$, complete the square to obtain the Apollonius circleIts centre is $(a(1+{\lambda }^2)/(1-{\lambda }^2),0)$ and its radius is $2a\lambda /|1-{\lambda }^2|$. The centres are positive for $\lambda <1$ and negative for $\lambda >1$.

At the allowed exceptional value $\lambda =1$, the locus is the line $x=0$, the perpendicular bisector of the two fixed points. It is a generalized circle, but not an ordinary finite-radius circle. The point $z=-a$ is excluded from the original ratio.