<s>[INST] The intersection of a sphere and the xy-plane is the circle $(x-1)^2+(y-2)^2=5$ and point $(1,2,1)$ is on the sphere The intersection of a sphere and the $xy$-plane is the circle $$(x-1)^2+(y-2)^2=5$$ and point $(1,2,1)$ is on the sphere. Find the center and radius of the sphere.
When the sphere and $xy$-plane intersect than $z=0$. 
I tried $(x-1)^2+(y-2)^2+(z+?)^2=5$ but what can I do next? 
 [/INST] Let $O(a,b,c)$ be center of sphere, $A(1,2,0)$ be the center of intersection circle, $B(1,2+\sqrt{5},0)$ be the point of the sphere and intersection circle, $C(1,2,1)$ be the point on the sphere.
Then the radius of the sphere is $R=OC=OB=AO+AC=AO+1$. 
The triangle $AOB$ is right, so:
$$AO^2+AB^2=OB^2 \Rightarrow AO^2+5=(AO+1)^2 \Rightarrow AO=2.$$
Thus:
$$R=AO+1=3;O(1,2,-2).$$
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