Find the equations of planes that just touch the sphere (x−2)2+(y−4)2+(z−4)2=25 and are parallel to xy plan, the yz plane, and xz plane.?

A sphere given in the form (x–a)² + (x–b)² + (x–c)² = R²has 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[1]. . . the xy plane . . .[2]. . . the yz plane . . .[3]. . . the zx plane . . .then OP must be parallel to the[1]. . . z-axis . . .[2]. . . x-axis . . .[3]. . . y-axis . . .so that[1]. . . P = (a, b, c±R) = (2, 4, 4±5). . .[2]. . . P = (a±R, b, c) = (2±5, 4, 4) . . .[3]. . . P = (a, b±R, c) = (2, 4±5, 4) . . .and the tangent plane is consists of all points (x,y,z) such that[1]. . . z = c±R = 4±5. . .[2]. . . x = a±R = 2±5 . . .[3]. . . y = b±R = 4±5 . . . respectively.... Show More

center is (2,4,4), radius is 5required planes arez=9, z=-1x=-3, x=7y=-1, y=9... Show More