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Find. the distance from the sphere %%x^2+y^2+z^2 - 6z = -4%% to the %%xy%%-plane.
To find where this sphere is centered we can start by completing the square on the z terms. (This is assuming that the -6x was meant to be a -6z)
%%x^2 +y^2 + z^2 -6z + 9 -9= -4%%
%%x^2 + y^2 + (z-3)^2 -9= -4%%
Adding the constant term over we find that this is a sphere centered at (0,0,3) with a radius of %%sqrt(5)%%.
%%x^2 + y^2 + (z-3)^2 = 5%%
The point on the sphere closest to the %%xy%%-plane can be found by adding %%-sqrt(5)%% to the z value of the sphere's center point. Therefore the sphere is %%(3-sqrt(5))%% units away from the %%xy%%-plane.