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How to Find the Locus of Points

A locus in mathematics is a set of points that falls within a specified constraint. This constraint is often represented as a function or, in the case of geometry, the rules for a shape. Determining a locus of points requires proving two facts: that a set of points falls within the specified constraint; that every point that is specified in the constraint is shown. This process is first introduced in advanced geometry and later re-evaluated when examining the limits of functions in introductory calculus.

Instructions

    • 1

      Determine the constraint that must be followed. For example, a student may be asked to find "the locus of points that represents the circle." In this case, the underlying constraint would be that all points follow the formula for a circle.

    • 2

      Graph a smooth curve that encompasses all the points set forth by the constraint. From the example above, you would draw a circle.

    • 3

      Ensure that all points possible are shown in the representation of the constraint. In the case of a circle, a line drawn with equal radius around the center point encompasses all possible points and therefore fulfills the requirement that all possible points be represented.

    • 4

      Prove that every point falls within the constraint set forth previously. It is simple, when examining a circle, to state that the formula of the circle was used to generate the graph and therefore the locus of points presented represents a line that follows the constraint.

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