In the next section, we show two different ways to get an ellipse. There are actually three ways but we will stick with two for now. One way is to slice a cone with a plane. The other way you may be more familiar with from high school. You draw an ellipse with a piece of string and two thumbtacks.

So you may be asking why do those two different definitions define the same figure? I mean, they seem so different. Well, here is a great video that explains this. This is absolutely fascinating!

Figure 1
[Source: Wikipedia]

Figure 2
[Source: Wikipedia - Ellipse]

The standard equation for an ellipse is \(\displaystyle{ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 }\). Figure 2 contains more information than we need right now but it will suffice. The longer axis is called the major axis (in this plot it is horizontal). The shorter axis is called the minor axis. The vertices are located on the ellipse where it crosses the major axis. The foci are also on the major axis, labeled F₁ and F₂ on this plot.

The major axis is determined by the denominators, \(a^2\) and \(b^2\). The larger value is in the denominator of the major axis, i.e. if \(a > b\) then the major axis is parallel to the x-axis. We need to define a value c where \(c^2=\abs{a^2-b^2}\) which will help us determine the location of the foci.

These tables contain the main attributes of an ellipse. We assume here that \(a > b\). Similar equations exist for \(a < b\).

1. Since the foci are closer to the center than the vertices, it follows that \(c < a\) and therefore \(0 < e < 1\).
2. Notice in the standard form of the equation, both terms are positive. This is how you know the graph is an ellipse and not a hyperbola.
3. In the general form of the equation, \(Ax^2+Bxy+Cy^2+\) \(Dx+Ey+F=0\), \(A > 0\) and \(C > 0\).
4. The eccentricity e is not the same as the irrational constant \(e \approx 2.72\).

Okay, time for some fun videos about ellipses. Here are a couple of videos about playing pool on an elliptical table. They clearly show the relationship between the foci and demonstrate some fun physics at the same time.

Stand Up Math - Why is there no equation for the perimeter of an ellipse? [21min-4secs]

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