Locus

Many problems in coordinate geometry ask us to find the equation of a curve starting from a property that characterises its points. For example: which are all the points of the plane that are equidistant from the two endpoints of a segment? Those equidistant from a point and a line? Those at a fixed distance from a centre?

The answers — the perpendicular bisector of the segment, the parabola, the circle — are familiar curves, but it is instructive to derive them from their locus definition.

Definition — Locus

A locus is the set of all and only the points of the plane that enjoy a given property. The equation of the locus is the equation in $x,y$ that the points of the locus satisfy and that no other point of the plane satisfies.

The method for deriving its equation is always the same:

1. take a generic point $P(x;y)$;
2. translate the property into an equality between distances;
3. square the terms containing a root (always allowed, since distances are $\ge 0$);
4. simplify until an equation in $x$ and $y$ is obtained.

Links

Topics: Geometric loci
Concepts: Point-line distance · Distance between points · Locus · Cartesian plane · Line
Skills: Analytical geometry