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High School Precalculus: Help and Review32 chapters | 297 lessons

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will define an ellipse, its semi-major axis, and a few other characteristics of an ellipse. We will also see how to find the length of the semi-major axis of an ellipse and solve an application involving this process.

Are you familiar with the shape of an ellipse? Have you ever eaten a burrito? Wait, what? What do these two things have to do with each other? It just so happens that they do have something to do with each other! You see, the end of a burrito takes on the shape of an ellipse!

Notice that the end of the burrito looks like a circle that's been squashed down a bit? Well, that is exactly how an ellipse looks!

Of course, a squashed down circle is not the technical definition of an ellipse! To be more formal, we define an **ellipse** as the set of all points with distance from two fixed points, *F* and *G*, adding up to the same number. We call the two fixed points, *F* and *G*, the foci of the ellipse. Even more formally, we say that an ellipse is the set of all points, *A*, such that if *F* and *G* are the foci of the ellipse, then *AF* + *AG* is constant.

That's a pretty technical definition, but good to know even if we do prefer to just think of an ellipse as a circle that has been squished down a little.

Now that we know what an ellipse is, let's talk about some of its parts. In particular, we want to talk about the semi-major axis of an ellipse. However, to introduce the semi-major axis of an ellipse, we must first recognize the major axis of an ellipse!

If we were to place an ellipse on an *x**y*-axis, with the origin at the center of the ellipse, one of the axes inside the ellipse would be a bit longer than the other, depending on if the circle was squished vertically or horizontally to create the ellipse.

Looking at an ellipse in this way, we call the two axes inside the ellipse the **major axis** and the **minor axis** of the ellipse. The major axis is the longer axis, and the minor axis is the shorter axis.

We can now define the **semi-major axis** of an ellipse as half of the major axis of an ellipse!

So far, so good! Nothing too complicated! Let's keep that going!

Now that we know what the semi-major axis of an ellipse is, let's talk about finding its length! We know that it is half of the major axis, so its length would be half of the length of the major axis, and as it turns out, the length of the major axis can be found by simply adding up the distance from one foci, *F*, to any point, *A*, on the ellipse and the distance from the other foci, *G*, to the same point *A*.

We see that *AF* + *AG* is equal to the length of the major axis. Therefore, we have the following;

Length of the major axis = *AF* + *AG*, where *F* and *G* are the foci of the ellipse, and *A* is any point on the ellipse.

Well, if that is the length of the major axis, then to find the length of the semi-major axis, we simply split it in half, or divide it by two!

Length of the semi-major axis = (*AF* + *AG*) / 2, where *F* and *G* are the foci of the ellipse, and *A* is any point on the ellipse.

That's pretty easy! Let's try putting this formula into action!

Suppose you have an elliptical shaped pond in your backyard, and you want to build a dock that goes out to the center of the pond from the edge farthest from the center. In other words, the dock represents the semi-major axis of the pond. A professional you hired a while back took some measurements from the foci of the pond to a point on the edge of the pond and found that the two distances are 32 feet and 16 feet.

You want to use this information to find the length that the dock will be. No problem! All we have to do is add up the two distances that the professional found to find the length of the major axis and divide this by 2!

Length of dock = (32 + 16) / 2 = 48 / 2 = 24

You find that the dock will be 24 feet. Problem solved!

An **ellipse** takes on the shape of a circle that has been squished horizontally or vertically. Technically, if *F* and *G* are the foci, then an ellipse is the set of all points, *A*, such that *AF* + *AG* is constant. An ellipse has two axes within it. The longer axes is the **major axis**, and it runs through the center of the ellipse from one end to the other at the widest part of the ellipse. The **semi-major axis** is half of the major axis. To find the length of the semi-major axis, we can use the following formula;

Length of the semi-major axis = (*AF* + *AG*) / 2, where *A* is any point on the ellipse, and *F* and *G* are the foci of the ellipse.

It is good that we are more familiar with this characteristic of an ellipse, since ellipses show up quite often in the world around us in architecture, landscaping, engineering, and even in a burrito! Speaking of which, I don't know about you, but I haven't stopped craving a burrito since that opening example! Time to eat!

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High School Precalculus: Help and Review32 chapters | 297 lessons

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