Find the equations of planes that just touch the sphere (x-2)^2 + (y-4)^2 +(z-4)^2 = 25 and are parallel to the xy plane, yz plane and the xz plane
A sphere given in the form
(x–a)² + (x–b)² + (x–c)² = R²
has center O = (a,b,c) and radius R. In our case
(a,b,c,R) = (2,4,4,5).
Any point P of this sphere whose tangent plane is parallel to some plane T must lie on a line perpendicular to T which passes through O (because radius OP has to be perpendicular to the tangent plane). Thus if the tangent plane at P is parallel to
. . . the xy plane . . .
. . . the yz plane . . .
. . . the zx plane . . .
then OP must be parallel to the
. . . z-axis . . .
. . . x-axis . . .
. . . y-axis . . .
. . . P = (a, b, c±R) = (2, 4, 4±5). . .
. . . P = (a±R, b, c) = (2±5, 4, 4) . . .
. . . P = (a, b±R, c) = (2, 4±5, 4) . . .
and the tangent plane is consists of all points (x,y,z) such that
. . . z = c±R = 4±5. . .
. . . x = a±R = 2±5 . . .
. . . y = b±R = 4±5 . . .
center is (2,4,4), radius is 5
required planes are