Show that the ellipsoid 3x2 +2y2 +z2 = 9 and the sphere x2 +y2 +z2 −8x−6y −8z +24 = 0are tangent to each other at the point (1, 1, 2)

Question

Show that the ellipsoid 3x2 +2y2 +z2 = 9 and the sphere x2 +y2 +z2 −8x−6y −8z +24 = 0are tangent to each other at the point (1, 1, 2)
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Solution 1

To show that the ellipsoid and the sphere are tangent at the point (1,1,2), we need to show two things:

  1. The point (1,1,2) lies on both surfaces.
  2. The gradients of the two surfaces at the point (1,1,2) are proportional.

Step 1: Check if the point (1,1,2) lies on both surfaces.

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