Definition and Consistency of a Locus

A locus is the set of all points in a plane that satisfy one specific geometric condition, and only that condition. Two ideas sit inside this one sentence, and both are needed for a full understanding. First, a locus is usually not a single point but an entire curve or region -- a point $P$ is imagined to move, obeying some rule (an equal distance from two fixed points, a fixed distance from a line, a constant ratio of two distances, and so on), and as it moves it sweeps out a shape: a straight line, a circle, a parabola, or some other curve. Second, and more subtle, is the requirement of consistency. An equation $f(x,y) = 0$ is said to represent the locus of $P$ under a condition $\mathcal{C}$ only when both of the following hold together: (i) every point whose coordinates satisfy $f(x,y)=0$ also satisfies the original geometric condition $\mathcal{C}$, and (ii) every point that satisfies $\mathcal{C}$ has coordinates that satisfy $f(x,y)=0$. In other words, the algebraic equation and the geometric condition must describe exactly the same set of points -- neither one may include a point the other excludes.

This is why, after simplifying an equation obtained from a geometric condition, the book insists on checking the converse: it is not enough that the condition leads to the equation; one must also confirm that the equation, read backwards, leads back to the condition. Skipping this check is a common source of error, particularly when the algebra involves squaring both sides of an equation (which can silently introduce solutions that do not actually satisfy the original, unsquared condition) or when a ratio condition is converted to a product of cross terms.

A locus problem in this chapter therefore always has two halves: setting up and simplifying an equation from a stated condition, and confirming that the simplification has not enlarged or shrunk the actual set of points. Once this idea of consistency is secure, the whole chapter becomes a single repeated exercise: translate a geometric sentence about a moving point into algebra, simplify it, and verify it -- the exact five-step method developed next.