Locus of a Point

A point is not a physical object but a position — indicated on paper by a dot, and located algebraically (once a coordinate system is fixed) by a unique ordered pair of real numbers $(x,y)$. The horizontal reference line is the $x$-axis, the vertical one the $y$-axis, and their meeting point is the origin. For a point $P$, $x$ is the (signed) distance from the $y$-axis and $y$ the (signed) distance from the $x$-axis; $x$ negative means $P$ is to the left of the $y$-axis, $y$ negative means $P$ is below the $x$-axis. (Applications sometimes use other letters than $x,y$, and different horizontal/vertical scales — the underlying idea is unchanged.)

Definition. The path traced out by a point moving under a stated geometric condition is called the locus of the point (plural: loci); equivalently, if an equation in $x,y$ is satisfied by infinitely many pairs $(x,y)$ (each a real solution of the equation), then the collection of the graphs of all those solutions is the locus of the equation.

Loci show up everywhere: the path of a cricket ball, replayed to judge an LBW appeal (third-umpire technology such as Hawk-Eye projects this locus forward through the batsman's legs); the cycloid traced by a fixed point on the rim of a wheel rolling without slipping along a straight line; the paths of missiles tracked during military engagements (e.g. Scud vs. Patriot missiles in the 1990–91 Gulf War) or of a satellite/space shuttle during launch. Three especially standard loci: a point equidistant from two fixed points $A,B$ traces the perpendicular bisector of $AB$; a point equidistant from two fixed lines $ox,oy$ traces the bisector of the angle $\angle xoy$; a point at fixed distance $r$ from a fixed point $O$ traces a circle of radius $r$.

Procedure for finding the equation of a locus.

1. Give the moving point $P$ (whose locus is sought) coordinates $(h,k)$.
2. Express the stated geometric condition(s) as equation(s) in $h,k$, any known quantities, and any unknown parameter(s).
3. Eliminate the parameter(s) so that only $h,k$ and known quantities remain.
4. Replace $h$ by $x$ and $k$ by $y$ — the resulting equation in $x,y$ is the required equation of the locus. …