Finding the tangent plane equation of surface x2 + 2Y2 + 3z2 = 21 parallel to plane x + 4Y + 6Z = 1

Finding the tangent plane equation of surface x2 + 2Y2 + 3z2 = 21 parallel to plane x + 4Y + 6Z = 1

Let the tangent plane be x + 4Y + 6Z = C (C is a parameter)
Then the normal vector is {1,4,6}
The normal vector of any point (x0, Y0, Z0) on the surface x2 + 2Y2 + 3z2 = 21 is {2x0,4y0,6}
Let the tangent point be (x, y, z)
So {1,4,6} = {2x, 4Y, 6}
The solution is x = 0.5, y = 1
The surface equation leads to Z = plus or minus 5 / 2
(0.5,1,2.5) and (0.5,1, - 2.5) are introduced into the tangent plane equation respectively
The solution is x + 4Y + 6Z = 19.5 and X + 4Y + 6Z = - 10.5