Conics: What is an ellipse?

Ellipses: (almost) everything you need to know

Informal definition

You may have heard of the definition that an ellipse is an oval. You can't answer this with "an oval is an ellipse", because that is circular logic. We can, however, say that an ellipse is a stretched-out circle. We take two diameters of a given circle that are perpendicular to each other, and scale each by any positive real number (preserving the midpoints). Thus we have a stretched-out circle, or an ellipse.

Two formal definitions

An ellipse can be defined in two ways. One is that it is any intersection of a plane with a double cone that is bounded. (Hyperbolas, parabolas, and the union of two lines are the unbounded intersections of a plane and a double cone). This allows for two special cases: a circle and a single point (which itself can be considered a degenerate case of a circle).

The second definition does not require leaving from the plane. Given two points in the plane (called foci, and denoted $F_1$ and $F_2$), and a positive real number (called the sum of focal radii, and denoted $2a$ - you'll see why it's $2a$ and not just $a$) that is greater than the distance between the two points, an ellipse is the set of all points $P$ such that $PF_1 + PF_2 = 2a$.

Equation of an ellipse with foci on the $x-$ axis and equidistant from the origin

We consider the simplest case: the foci are on the $x-$ axis and the midpoint of $F_1F_2$ is $(0,0)$, so the foci have coordinates $(\pm c, 0)$. We apply the distance formula:

$$\sqrt{(x-c)^2 + y^2} + \sqrt{(x+c)^2 + y^2} = 2a$$ $$\sqrt{(x+c)^2 + y^2} = 2a - \sqrt{(x-c)^2 + y^2}$$ $$(x+c)^2 + y^2 = 4a^2 - 4a\sqrt{(x-c)^2 + y^2} + (x-c)^2 + y^2$$ $$x^2 + 2cx + c^2 + y^2 = 4a^2 - 4a\sqrt{(x-c)^2 + y^2} + x^2 - 2cx + c^2 + y^2$$ $$4cx - 4a^2 = - 4a\sqrt{(x-c)^2 + y^2}$$ $$-\frac{c}{a}x + a = \sqrt{(x-c)^2 + y^2}$$ $$\frac{c^2}{a^2}x^2 - 2cx + a^2 = x^2 - 2cx + c^2 + y^2$$ $$\frac{a^2-c^2}{a^2}x^2 + y^2 = a^2 - c^2$$ $$\frac{x^2}{a^2} + \frac{y^2}{a^2-c^2} = 1$$

The last step works because of the fact that we have $2a>2c>0$, so therfore $a^2-c^2 \ne 0$. Now, if we make the substitution $b^2 = a^2 - c^2$, the familiar formula

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

arises.

Properties of an ellipse

In an ellipse, $2a$ is the length of the major axis and $2b$ is the minor axis. The distance beween the foci is $2c$, and $a,b,c$ satisfy $b^2+c^2=a^2$. The chord perpendicular to the major axis at one of the foci is called the latus rectum, or the focal chord. It has length equal to $\frac{2b^2}{a}$ (this can be shown easily by finding twice the positive $y-$ value associated with the $x-$ value $c$.

More general equations

If you instead have the foci on the $y-$ axis, then the distance formula yields an identical formula except $x$ and $y$ are switched.

If the midpoint of $F_1F_2$ were to be translated to $(h,k)$ instead of $(0,0)$, then $x$ and $y$ would be replaced by $x-h$ and $y-k$ respectively.

If $F_1F_2$ is not parallel to one of the axes, then the equation gets a bit more challenging. First of all, an $xy$ term will appear. The general equation for a conic section is $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$. Not all of these are ellipses, obviously - in the case where $B=C=D=F=0$, $A=1$, $E=-1$, you have $y=x^2$, which is obviously not an ellipse. In fact, a general conic section is an ellipse if and only if $B^2-4AC$ (here also called the discriminant) is less than $0$.

Special Cases

Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ describes an ellipse. Now, if $B=0$ and $A=C$, this is still an ellipse ($B^2-4AC = 0-4A^2<0$, but it's a special ellipse - it's a circle. In this case the distance between the foci is $0$, so the length of the minor axis is equal to that of the major axis.

A degenerate case appears when you have a circle, $x^2 + y^2 + ax + by + c = 0$, but specifically when $c = \frac{a^2 + b^2}{4}$. In this case, the square is already completed - after combining terms you get $(x+\frac{a}{2})^2 + (y+\frac{b}{2})^2=0$. In this case, the conic is a circle with radius $0$, or a single point. This is a degenerate conic section and is often not listed by textbooks to be one of the cases of conic sections.

Ellipses have two definitions that seem completely unrelated.

1. An ellipse is a dilated circle. This is the more intuitive definition. Ellipses look just like circles, but stretched out a bit. You could create such a thing by drawing a circle on a rubbery sheet and then stretching out the sheet in one direction perfectly evenly, but that's somewhat difficult to pull off. Another way to accomplish the same thing is to take a cylinder and slice through it at an angle, like so:

The cross-section of the cylinder produced by the slice is an ellipse.

2. An ellipse is a curve on a plane that surrounds two focal points such that the sum of the distances to the two focal points is a constant ratio for every point on the curve. To make an ellipse in this fashion, stick two pins into your paper, then wrap a loop of string around them. Take a pencil and trace around the inside of the loop. Because the length of the string doesn't change as you draw, the resulting curve has an interesting property: for every point you choose on the curve, the sum of the distances from that point to the two pins is constant.

This curve is an ellipse.

Let's think about how both can be called ellipses. One thing is created by slicing a cylinder, and the other is created by tracing around two pins with a loop of string. In 1822, the Belgian mathematician Germinal Pierre Dandelin discovered a really nice proof showing that these two objects are one and the same.

Start by slicing a cylinder, creating an ellipse according to the first definition. Next, drop two spheres into the cylinder, above and below the ellipse. These spheres have the same diameter as the cylinder, so they fit perfectly snugly and are tangent to the cylinder at their equators. Dandelin realized that the two points at which the spheres touch the ellipse are foci!

The reasoning is this: pick a point on the ellipse. The line segment from that point to the upper focus is tangent to the blue sphere, and the line segment from that point going directly up to the blue sphere's equator is also tangent to the blue sphere. Because the segments share a common endpoint and are tangent to the same sphere, they must have the same length. This argument is also true for the lower focus and the red sphere. The animation shows that regardless of where you are on the ellipse, the two blue segments are equal and the two red segments are equal.

But the sum of a red segment and a blue segment is constant, because it's the distance between the equators of the two spheres! Therefore, the ellipse satisfies the second definition, with the points at which the spheres and the ellipse touch being the foci.

(Video) Ellipse | Don't Memorise

by Don't Memorise

Watch this video by Don't Memorise to learn what an ellipse is!

Like the video says, an ellipse is the collection of points on a plane where the sum of the distances of each point from two fixed points is constant.

The foci are two fixed points on a plane. The sum of the distances of each point to the two foci must be greater than the distance between the foci.

Interestingly, a circle is an ellipse specifically when the foci coincide.

This video does a wonderful job talking about the applications of ellipses - I would encourage you to marvel at how simple shapes can have amazing properties!

Next, learn about the standard from of an ellipse!