Apollonius sphere ($|\mathbf{x}-\mathbf{a}|=\lambda |\mathbf{x}-\mathbf{b}|$)

For distinct points and $0<\lambda <1$, the distance-ratio locus is a sphere with center $(\mathbf{a}-{\lambda }^2\mathbf{b})/(1-{\lambda }^2)$ and radius $\lambda |\mathbf{a}-\mathbf{b}|/(1-{\lambda }^2)$. Completing the square proves this formula. It encloses $\mathbf{a}$ but excludes $\mathbf{b}$, because their distances to the center are $\lambda R$ and $R/\lambda$. It is the zero-potential locus of two opposite-sign point charges with magnitude ratio $\lambda$. Equal magnitudes instead give a plane.