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Shortest & largest distance from origo to ellipse

  1. Aug 13, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the largest and shortest distance from origo to the ellipse.

    2. Relevant equations
    Ellipse: [itex]g(x, y) = 13x^2 + 13y^2 + 10xy = 72[/itex]
    Function to optimize: [itex]F(x, y) = \sqrt{x^2 + y^2} [/itex]
    But this is easier to optimize: [itex]f(x, y) = x^2 + y^2 [/itex]

    3. The attempt at a solution
    I set up the equations, [itex]\nabla f = \lambda \nabla g[/itex], which got me that [itex]x = y[/itex]. [itex]g(x) = 36x^2 = 72[/itex] and [itex]x^2 = 2[/itex] which got me that one of the values (either minimum or maximum) is [itex]F(x, y) = 2[/itex].

    The question is, how do I get the other value?
  2. jcsd
  3. Aug 13, 2011 #2


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    If [itex]x^2= 2[/itex] then [itex]x= \pm\sqrt{2}[/itex]. Put that back into the equation of the ellipse and solve for y.
  4. Aug 13, 2011 #3
    Yes, but I got [itex]x^2 = 2[/itex] from that equation, only because [itex]x = y[/itex].
    I meant the other optima. According to the answers, 2 is the minimum. So I'm looking for the maximum.
  5. Aug 13, 2011 #4
    Ohh, nevermind... now that I checked the answer, [itex]x = -y[/itex].
    The thing is, I got [itex]x^2 = y^2[/itex] and assumed that [itex]x = y[/itex], but obviously [itex]x = -y[/itex] is correct too and gives different points.
    Thanks anyway :).
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