= Apollonius sphere
{c}
{title2=$|\mathbf x-\mathbf a|=\lambda|\mathbf x-\mathbf b|$}
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.
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