# Conic sections: Can focal points be outside the ellipse?

1. Sep 26, 2012

### Bipolarity

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 $x^{2} + \frac{y^{2}}{4} = 1$ are $(-\sqrt{3},0) (\sqrt{3},0))$.

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

BiP

2. Sep 26, 2012

### lavinia

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. Sep 27, 2012

### HallsofIvy

Yes, that's wrong. If an ellipse is given by
$$\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1$$
with b> a, then the foci are at (0, c) and (0, -c) with $c^2= a^2- b^2$
so the foci of this ellipse are on the y-axis, not the x-axis.

4. Sep 27, 2012

### Ben Niehoff

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. Oct 11, 2012

### HallsofIvy

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.