Conic sections: Can focal points be outside the ellipse?

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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
 

Answers and Replies

  • #2
lavinia
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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
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.
 
  • #3
HallsofIvy
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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
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.
 
  • #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.
 
  • #5
HallsofIvy
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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.
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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