Show that the plane `2x-2y+z+12=0` touches the sphere `x^2+y^2+z^2-2x-4+2z-3=0.`

Explanation

The given plane will touch the give sphere if the perpendicular distance from the centre of the sphere to the plane is equal to the radius of the sphere. The centre of the given sphere `x^(2)+y^(2)+z^(2)-2x-4y+2z-3=0` is (1, 2, -1) and its radius is `sqrt(1^(2)+2^(2)+(-1)^(2)-(-3))=3`.
Length of the perpendicular from (1, 2, -1) to the plane `2x-2y+z+12=0` is
`" "|(2(1)-2(2)+(-1)+12)/(sqrt(2^(2)+(-2)^(2)+1^(2)))|=(9)/(3)=3`
Thus, the given plane touches the given sphere.