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Lagrange on an Ellipse to find Max/Min Distance

  1. Nov 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Bci0M.png


    2. Relevant equations

    Lagrance Multipliers.

    3. The attempt at a solution

    This is a pretty dumb question, and I feel a little embarassed asking but..

    I know how to do the Lagrange part (I think). I'm assuming you maximize/minimize the distance, [itex]\sqrt{x^{2} + y^{2} + z^{2}}[/itex] subject to the constraints of the ellipse.


    The part I'm having trouble with is finding the constraints of the ellipse - could someone help point me in the right direction?
     
  2. jcsd
  3. Nov 29, 2012 #2

    Ray Vickson

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    It is a lot easier to maximize or minimize distance-squared; do you see why it is OK to do that? As for the constraints, ask yourself: how can I tell if a given point (x,y,z) is on the ellipse? What are the conditions for that?
     
  4. Nov 29, 2012 #3
    Ah yes, the thought of it passed my mind, but I didn't think it was viable until you pointed it out - thanks! :)

    As for the constraints, I know the equations of the cylinder and the plane that create it.. but when I try to put them together (i.e., [itex]x^{2} + y^{2} = x + y + z[/itex]), it doesn't seem to give me the correct equation. I might just be having a brain fart, but I just don't know what I'm missing or doing incorrectly here :(
     
  5. Nov 29, 2012 #4
    Just gave it an attempt, and my maximization problem became:


    Maximize [itex]x^{2} + y^{2} + z^{2}[/itex] subject to [itex]x + y - \sqrt{1 - x^{2}} - \sqrt{1 - y^{2}} = 0[/itex]

    Is this correct? If so, do I need to substitute out z in terms of x and y? Because I have a feeling that would make things very complicated..
     
  6. Nov 29, 2012 #5

    Ray Vickson

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    You wrote two equations, not one, so you need two Lagrange multipliers.
     
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