A set in a disc

Question

Let $z_1,z_2$ be any two given complex numbers and $r$ be a nonnegative real number. I am trying to find a smallest closed disc containing the set $\left\{z\,\big|\,\left|\frac{z-z_1}{z-z_2}\right|\leq r\right\}.$ May I seek your help in proceeding towards finding the center and radius of such a disc?

Answer 1

Assuming $r\neq1$. This is the circles of Apollonius. We have two ends of a diameter on the line through $z_1$ and $z_2$, given by $PP_1:PP_2=1:\pm r$ ($P_i$ is the point with coordinate $z_i$), i.e., $$ \left(\frac{z_1+rz_2}{1+r}\right)\text{ and }\left(\frac{z_1-rz_2}{1-r}\right) $$ so that gives you the centre = $\frac{z_1-r^2z_2}{1-r^2}$ and radius = $\left|\frac{z_1-z_2}{1-r^2}\right|r.$