Determine the tangent planes of the level surface given by x^2+y^2+x^2 = 1 at the points (x,y,0) and show that they are all parallel to the z-axis.

here is my working thus far:

f(x,y,z) = x^2+y^2+z^2-1

then the tangent plane equation is:

(∂f/dx)|(x,y,0)(x-x) + (∂f/dy)|(x,y,0)(y-y) + (∂f/dz)|(x,y,0)(z-0) = 0

the first two terms go to zero since (x-x = y-y = 0), and the last term leaves 0z=0 (no information!)

Mathematically, where is my problem?