The floor of the banquet hall in a hotel is made of polished stone. There is a large chandelier attached to the ceiling of the hall. Consider floor of the hotel as a plane having the equation x - y + z = 4 and chandelier is suspended at the point (1, 0, 1) from the wall.

On the basis of above information, answer the following questions.

The equation of the plane in the banquet hall parallel to the x - y + z = 4, and at a unit distance from the point (1, 0, 1), is : 

$x-y+z=2-\sqrt{3}$

$\text{Given plane: }x-y+z=4.$

$\text{Any plane parallel to it: }x-y+z+d=0.$

$\text{Distance from }(1,0,1)\text{ to plane }x-y+z+d=0 \text{ is }1.$

$\frac{|1-0+1+d|}{\sqrt{1^2+(-1)^2+1^2}}=1.$

$\frac{|2+d|}{\sqrt{3}}=1.$

$|2+d|=\sqrt{3}.$

$d=-2\pm\sqrt{3}.$

$\text{Plane: }x-y+z-2\pm\sqrt{3}=0.$

$\text{Required planes: }x-y+z-2+\sqrt{3}=0 \text{ or } x-y+z-2-\sqrt{3}=0.$