# Locus of Points: Definition, Methods & Examples

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• 0:00 What Is a Locus of Points?
• 1:09 Locus of Points and Equations
• 2:09 More Examples of Locus…
• 4:39 Lesson Summary
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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 will define a locus of points. We will look at various examples of a locus of points, and we will look at an example of finding an equation representing a locus of points in a real-world setting.

## What is a Locus of Points?

We're going on a hiking trip. We decide that we want to hike 5 miles out from and 5 miles back to our starting point, and we're trying to decide where in the hiking area we would like the turn-around point to be. We look at a map of the area and plot out all the points that are 5 miles from our starting point.

You notice that one of the routes ends up at a scenic lookout. This route will make for a beautiful hike. Let's get going.

It may surprise you that the circle you created on the map actually has some mathematical significance, and it's not just because it's a circle. As we said, this circle is a set of points that are all 5 miles from our starting point. In mathematics, we call this a locus of points.

A locus of points is a set of points that satisfy a certain property or condition. The circle on the map is the set of points satisfying the property that they're all 5 miles from our starting point, so they form a locus of points. You probably didn't realize you were creating a mathematical set of points when we were mapping out our route.

## Locus of Points and Equations

A locus of points usually results in a curve or surface. For instance, in our hiking example, the locus of points 5 miles from our starting point resulted in a curve that's a circle. Now, how do we usually represent curves algebraically? If you're thinking we use an equation, you're exactly right.

It turns out that the solutions to an equation are an example of a locus of points, because those solutions are a set of points that satisfy the property that they make the equation true. For example, if we let our starting point be the origin on a coordinate system, then the set of all points that are 5 miles from that point make up the circle that's represented by the equation x2 + y2 = 25.

Therefore, our locus of points can be described as the set of points that are 5 miles from our starting point, or as the set of points that satisfy the equation x2 + y2 = 25, if we let our starting point be the point (0,0).

## More Examples of Locus of Points

Because a locus of points is simply a set of points that satisfy some conditions or properties, there are a lot of different examples of this mathematical concept. Let's take a look at some more examples of these so that we can really solidify our understanding of the concept.

You may be familiar with an ellipse. An ellipse is a two-dimensional shape that looks like a circle that has been stretched a bit horizontally or vertically. However, what you may not know is that we can define an ellipse formally as a set of points, such that the sum of the distances from two points, called the foci, is the same for each point in the set.

We just defined an ellipse as a set of points that satisfy a certain property, so is this ringing any bells? Ah! Yes, an ellipse is a locus of points that satisfy the property that the sum of the distances from the point to each of the foci of the ellipse is constant. Pretty neat, huh?

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