Question

Show that the plane 3x + 12y - 6z = 17 touches the hyper boloid 3x2 - 6y2 + 9z2 + 17 = 0, and find the point of contact.

Answer 1

Given hyperboloid: x^2/3 - y^2/2 + z^2/1 = - 17/9

Tangential plane at (a, b, c): a x/3 – 2 by/3 + c z/1 = - 17/9 => - 3a/17 x + 6b/17 y – 9c z/17 = 1

Given plane: 3 x/17 + 12 y/17 - 6 z/17 = 1

Compare coefficients: a = - 1, b = 2, c = 2/3 Point of contact = (- 1, 2, 2/3).