The equation $xy = 0$ in three-dimensional space represents

Find the equation of a plane that makes equal intercepts of unit length on the axes.

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $x+y+z=1$.

If a plane meets the coordinate axes at $A, B$,and $C$ such that the centroid of the triangle $ABC$ is $(1, 2, 4)$,then the equation of the plane is:

The vector equation of a plane,which is at a distance of $8$ units from the origin and which is normal to the vector $\vec{n} = 2\hat{i} + \hat{j} + 2\hat{k}$,is

If the foot of the perpendicular from $(0,0,0)$ to the plane is $(1,2,2)$,then the equation of the plane is