Locus and Its Equation

In everyday language a path is the trail traced out by a moving object — the arc of a thrown ball, the circle swept by the tip of a clock hand. In coordinate geometry we make this idea precise. A locus (plural loci) is the set of all points in a plane that satisfy one stated geometric condition, and only those points. Every point that meets the condition belongs to the locus; every point that fails it is excluded.

The great advantage of using coordinates is that a geometric condition on a moving point $P(x, y)$ can be written as an algebraic equation in $x$ and $y$. That equation is called the equation of the locus. It has a two-way meaning:

The standard method to find the equation of a locus is always the same three steps:

1. Take a general point $P(x, y)$ on the locus (its coordinates are unknowns, deliberately left as $x$ and $y$).
2. Write the given geometric condition as a relation involving $P$ — usually using the distance formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ learnt earlier.
3. Simplify that relation until it is a clean equation in $x$ and $y$. Squaring both sides early removes the awkward square roots that the distance formula introduces.

Two loci occur so often that it is worth naming them in advance. If a point stays a fixed distance $r$ from a fixed point $C(a, b)$, its locus is a circle of radius $r$, with equation $(x-a)^2 + (y-b)^2 = r^2$. If a point stays equidistant from two fixed points $A$ and $B$, its locus is the perpendicular bisector of the segment $AB$ — a straight line. This second case is the natural bridge from locus to the rest of this chapter, which is entirely about the straight line.