All points that are 2 units away from ...

"Describe the set with one or two equations:
All points that lie 2 units from (0,0,1) and 2 units from (0,0,-1)."

I realize that it's a circle in the xy-plane, but I'd like to  make my reasoning a bit more solid than "It's obvious!"?

I've tried
   x^2 + y^2 + (z-1)^2 = x^2 + y^2 + (z+1)^2,
but it doesn't really get me anywhere.

Well the equations of the two spheres are

x^2+y^2+(z+1)^2 = 4
x^2+y^2+(z-1)^2 = 4

Now suppose a point (x,y,z) is on both.  Subtracting the second from the first gives

(z+1)^2-(z-1)^2 = 4z = 0

So z = 0 and the intersection is contained in the xy plane.  Setting z = 0 in either of the equations gives

x^2+y^2 = 3

So that's the equation of your circle.

>Setting z = 0 in either of the equations gives

Gah, of course. Thanks!