# How to Find the Major Axis of an Ellipse

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• 0:03 The Definition of an Ellipse
• 0:41 Finding the Major and…
• 3:00 Solutions to a Problem
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Lesson Transcript
Instructor: Laura Pennington

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

In this lesson, we'll review the definition of an ellipse, learn how to find the major axis of an ellipse given the foci and one point on the ellipse, and look at an application of this process.

## The Definition of an Ellipse

When we think of geometrical shapes, the word ellipse doesn't immediately come to mind, but ellipses are more common than we think. They can be very small, like a bird's egg, or very large, like the orbit of the earth around the sun. An ellipse is a geometrical shape that looks like a stretched out circle that's been pulled either horizontally or vertically. An ellipse can be further defined by naming two important points on the ellipse as the foci (the plural of focus), the points, F and G. Another point on the ellipse will be called P.

## Finding the Major and Minor Axes

Okay, get ready for some vocabulary! The point in the center of the ellipse is the center of the ellipse. That was easy! Next, we draw a line through the center connecting the foci on either end of the ellipse. The longest line is the major axis. The major axis connects F and G, which are the vertices, the points on the ellipse. If we draw a line that is 90 degrees, or perpendicular, to the major axis, the short line is the minor axis. The points that the minor axis connects are the co-vertices. It is a lot of vocabulary, but it's necessary for finding the major axis of an ellipse.

Since we know the foci F and G and another point on the ellipse, finding the major axis wasn't really that hard. Now here's an easy rule to remember about ellipses: Since FP + GP is constant for every point on the ellipse, it is also the same for any point, P and a vertex (the singular of vertices) as shown in the next diagram:

The equation of the length of the major axis would look like this:

FP + GP = FV + GV

FV and GV give us the length of the major axis. If F and G are the foci of an ellipse, and P is any point on the ellipse, then the major axis is also known as FP + GP.

Now that we know the foci and a point on the ellipse, we can follow three easy steps to find the major axis:

• Step 1: Use the distance formula to find FP
• Step 2: Use the distance formula to find GP
• Step 3: Add FP and GP

Pretty simple, huh? It's a good idea to review the distance formula since we 'll be using it to find distances. The distance formula used to find the distance between (x1, y1) and (x2, y2) is found in this image:

## Solutions to a Problem

Now, to find the major axis in the above image, all we need is the foci, F and G, and P, as the point on the ellipse. We then follow these three steps:

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