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Crazy lagrange problem

  1. Nov 4, 2004 #1
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    Ok this is the question I had on a test today:

    given this constraint equation z^2-xy+1=0 find the min. distance from the origin using Lagrange method.

    so basically you use D^2=x^2+y^2+z^2 as the other equation. however, it basically goes nuts from there. especially if you set it up like you are suppose to.
    Fx=(lambda)Gx
    Fy=(lambda)Gy
    Fz=(lambda)Gz
    g=0

    (capitals are partial derivatives)

    with f as the distance formual and g as the constraint

    this one sucks but if someone could help it would be greatly appreciated
     
  2. jcsd
  3. Nov 4, 2004 #2

    matt grime

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    Science Advisor
    Homework Helper

    Minimizing D is the same as minimizing D^2, so we solve

    2x = -ky
    2y=-kx
    2z=2kz

    whence k=1, x=-y, which leads to nonsense, or z=0, and 2x=-ky=k^2x, so k=sqrt(2), also note that xy=1, and the solution follows.

    edit, thanks to arildno, it should read:

    2x=-ky=k^2x/2, ie

    4x=xk^2, whence k=+/-2

    from which you should be able to get the answer.
     
    Last edited: Nov 4, 2004
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