Answer
Sphere with center $(x_0,x_0,z_0)$ and radius $1$.
Work Step by Step
$$|\lt x,y,z\gt-\lt x_0,x_0,z_0\gt|=1$$
$$|\lt x-x_0,y-y_0,z-z_0\gt|=1$$
$$\sqrt{( x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}=1$$
$$( x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}=1^{2}$$
The above equation represents the sphere with center $(x_0,x_0,z_0)$ and radius $1$.
