How many spheres go through two points and are tangent to a line

Question

How many spheres go through two points $A(1,3,1)$ and $B(3,2,2)$ and are tangent to line $Oz$? Find their radius.

I try to exploit the following identities $IA=IB=d(I,Oz)$ where $I$ is the center of the sphere, but still not find the exact equation or even its radius.

Answer 1

Consider a family of circles which are tangent to the $z$ axis and passing through the points $A$ and $B.$

Answer 2

There are infinitely many spheres passing through $A$, $B$ and tangent to the $z$ axis. Let $P=(0,0,a)$ the tangency point: the center $O$ of the sphere must then be on the plane $z=a$ and must have the same distance from $A$, $B$ and $P$.

Center $O$ is then the intersection between plane $z=a$ and the line perpendicular to plane $ABP$ and passing through the circumcenter of triangle $ABP$. See here a construction of the sphere made with GeoGebra.

The center can be readily found: if $O=(x,y,a)$ then you just have to solve the system $$ \cases{ x^2+y^2=(x-1)^2+(y-3)^2+(a-1)^2\\ x^2+y^2=(x-3)^2+(y-2)^2+(a-2)^2\\ } $$ to find $x$ and $y$.