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Conic sections: Can focal points be outside the ellipse?

  1. Sep 26, 2012 #1
    Can an ellipse's focal points be outside the ellipse? I have tried googling this, but without any good explanations or answers.

    According to my calculations, the focal points of the ellipse defined by [itex] x^{2} + \frac{y^{2}}{4} = 1 [/itex] are [itex] (-\sqrt{3},0) (\sqrt{3},0)) [/itex].

    I maybe wrong of course, but does this mean that the focal points of an ellipse can indeed be outside the ellipse?

    BiP
     
  2. jcsd
  3. Sep 26, 2012 #2

    lavinia

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    The focal points of an ellipse are always interior. In a plane, an ellipse is the set of points that are equidistant to two points.
     
  4. Sep 27, 2012 #3

    HallsofIvy

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    Yes, that's wrong. If an ellipse is given by
    [tex]\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1[/tex]
    with b> a, then the foci are at (0, c) and (0, -c) with [itex]c^2= a^2- b^2[/itex]
    so the foci of this ellipse are on the y-axis, not the x-axis.
     
  5. Sep 27, 2012 #4

    Ben Niehoff

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    In the limit as the eccentricity goes to zero, an ellipse becomes a circle; both focal points converge to the center. In the limit as the eccentricity goes to infinity, an ellipse becomes a line segment, where the focal points are at the endpoints. In between, the focal points are always inside the ellipse.
     
  6. Oct 11, 2012 #5

    HallsofIvy

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    The eccentricity of an ellipse is always between 0 and 1 so it cannot "go to infinity". As the distance between foci goes to infinity, the eccentricity goes to 1.

    Eccentricity 1 gives a parabola, eccentricity greater than 1 is a hyperbola.
     
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