Algebra/Chapter 17/Loci

Algebra

The Distance Formula

Algebra Chapter 17: Conic Sections Section 2: Loci of Points

Conic Sections

17.2: Loci of Points

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Up to this point, all of the equations that we have solved or graphed had only one variable in mind.

Loci

A locus (plural: loci) is one of a set of points which satisfies a set of conditions. The usual result is a curve or a surface.

In real life, you likely heard of an object's location. As a matter of fact, the word "location" comes from the word "locus" itself. Loci define where in a plane or space that an object is located.

The photo on the right illustrates a set of points (or locus) of the headlights as traced under the condition that all vehicles follow the path of the road. In this analogy:

Set of Points: The headlight's locations as seen in the photograph
Condition: The road that the vehicles must follow
Locus: The lanes in the road as illustrated

Unfortunately, there is no general equation for finding loci. However, the following steps are typically used to determine the equation for a locus.

1. Contruct a diagram showing the given information
2. Locate several points that satisfy the rule or condition
3. Draw a curve or line using the located points
4. Write the equation

Equidistance

Example 17.1: Find the locus of points $P$ such that $P$ is equidistant from both axes. A point on the x-axis has the coordinates $(x,0)$, and a point on the y-axis has the coordinates $(0,y)$.