What is the shortest distance between the spherical surfaces $(x-1{)}^2+(y+2{)}^2+(z-3{)}^2=7$ and

$(x+3{)}^2+(y-2{)}^2+(z-1{)}^2=17?$

Remember that two spherical surfoces can have one of several possible geometric arrangements:

Eoch surface could be completely outside the other $($eg$.$ two marbles not touching$)$

One surfoce could be completely inside the other $($e$.$g$.$ a bubble floating freely inside another bubble$)$

The two spheres could intersect at a single point $($e$.$g$.$ two marbles touching eoch other or a marble inside and

resting against the inner surface of a ping pong ball$)$

The two spheres could partially overlap $($e$.$g$.$ sound waves in air from two people clapping after each has heard

the other$)$

The two spheres could coincide, which hoppens if and only if the two equations represent the same sphere

In the first two coses there is a smallest positive distance between the surfoces. In the others there is not. Correct

classification of the geometry is key!