How do you find the equation of a plane that is tangent to both the sphere w/ radius 3 and center (1,1,1) and the sphere w/ radius 4 and center (2,3,4)?

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Answer 1 · 2016-11-06T23:10:14

Please see the explanation for steps leading to the equation:
#2x + 4y + 6z = 19#

The equation of the first sphere is:

#(x - 1)^2 + (y - 1)^2 + (z - 1)^2 = 3^2#

Expanding the squares:

#x^2 - 2x + 1 + y^2 - 2y + 1 + z^2 - 2x + 1 = 9#

Subtract 3 from both sides:

#x^2 - 2x + y^2 - 2y + z^2 - 2x = 6##" [1]"#

The equation of the second sphere is:

#(x - 2)^2 + (y - 3)^2 + (z - 4)^2 = 4^2#

Expand the squares:

#x^2 - 4x + 4 + y^2 - 6y + 9 + z^2 - 8z + 16 = 16#

Subtract 29 from both sides:

#x^2 - 4x + y^2 - 6y + z^2 - 8z = -13##" [2]"#

Subtract equation [2] from equation [1]:

#2x + 4y + 6z = 19#