Parabola is an ellipse, but with one focal point at infinity While I was reading about conic sections, I came across the following statement:


  A parabola is an ellipse, but with one focal point at infinity.


But it is not clear to me. Can someone explain it clearly?
The equation for an ellipse with a focus at $(0,0)$ and the other at $(0,2ae)$ keeping $a(1-e)=f$ (where $f$ is distance from the vertex to the focus of the ellipse, which ends up being the focal length of the parabola) is
$$
\frac{x^2}{a^2(1-e^2)}+\frac{(y-ae)^2}{a^2}=1
$$
which is equivalent to
$$
\frac{x^2}{f(1+e)}+\frac{y^2-2aey}{a}=f(1+e)
$$
If we let $a\to\infty$ (and therefore $e=1-\frac fa\to1$), we get
$$
y=\frac{x^2}{4f}-f
$$
which is a parabola.

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Think of a pair of cones with vertical axes and identical cone-angles, one facing up and one facing down. Place them so they're touching at one point.  I'll call this a "cone" for now. 

If you slice this object with a plane perpendicular to the axis, you'll get a circle of some radius, or perhaps a single point, which you could call a circle of radius 0. 

If you slice it with a slightly tilted plane, you'll get an ellipse (or a single point). Thus circules and ellipses are both "cross-sections" of a cone, or "conic sections". 

If you tilt the slicing plane further, so it's nearly vertical, it'll intersect both cones, resulting in a hyperbola, or, if the plane passes through the cone-point, a pair of intersecting lines. So intersecting lines are kind of a "limit" of hyperbolas. 

Go back to ellipses: place the plane to slice an ellipse from the upper half of the cone.  Tilt your slicing plane more and more, making the ellipse more and more eccentric. There's a last possible amount of tilt before you start also slicing into the bottom part of the cone. At that tilt, the intersection is no longer an ellipse, but instead a parabola. 

So it's reasonable to say that a parabola is a limit of ellipses. 

Of course, if you tilt a tiny bit more, you start getting hyperbolas, so it's also reasonable to say a parabola is a limit of hyperbolas.