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.?

2 Answers

• 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

  [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.

• center is (2,4,4), radius is 5

  required planes are

  z=9, z=-1

  x=-3, x=7

  y=-1, y=9