= Solution
Squaring the ratio of moduli gives
$$
(x-a)^2+y^2=\lambda^2[(x+a)^2+y^2].
$$
For $\lambda\neq1$, complete the square to obtain the <Apollonius circle>
$$
\boxed{\left(x-\frac{a(1+\lambda^2)}{1-\lambda^2}\right)^2+y^2=\left(\frac{2a\lambda}{|1-\lambda^2|}\right)^2.}
$$
Its 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 $\lambda>1$.
\b[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.
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