find all the places on the hyperboloid x^2-y^2-z^2=-1 where the tangent plane is
ID: 3188525 • Letter: F
Question
find all the places on the hyperboloid x^2-y^2-z^2=-1 where the tangent plane is parallel to the plane z=x+y.Explanation / Answer
When z = 2, we have -x^2 - y^2 + 4 = 1 ==> x^2 + y^2 = 3. Moreover, the top half of the hyperboloid may be rewritten z = v(1 + x^2 + y^2). Therefore, using polar coordinates yields V = ??? 1 dV = ?(? = 0 to 2p) ?(r = 0 to v3) ?(z = v(1 + r^2) to 2) 1 (r dz dr d?). Evaluating this, we get 2p ?(r = 0 to v3) r (2 - v(1 + r^2)) dr = 2p ?(r = 0 to v3) (2r - r (1 + r^2)^(1/2)) dr = 2p [r^2 - (1/3)(1 + r^2)^(3/2)] {for r = 0 to v3} = 2p [(3 - 8/3) - (-1/3)] = 4p/3.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.