Equation of Locus

5.1.1 Equation of Locus

Suppose every point on a locus has coordinates satisfying some algebraic equation in $x$ and $y$, and — just as importantly — no point off the locus satisfies that equation. Then that equation is called the equation of the locus.

Worked Example 1. Every point on the X-axis has $y$-coordinate $0$, and this is true only for points on the X-axis (no off-axis point has $y=0$). So the equation of the X-axis is simply

Worked Example 2. Let $L = \{P \mid OP = 4\}$; find its equation. Take a general point $P(x,y)$ on $L$. Since $OP = 4$, we also have $OP^2 = 16$, so by the distance formula

This is the equation of the locus $L$, and geometrically the locus is a circle of radius 4 centred at the origin.

Worked Example 3. Find the equation of the locus of points equidistant from $A(-3,0)$ and $B(3,0)$, and identify it. Let $P(x,y)$ be any point on the required locus. Since $P$ is equidistant from $A$ and $B$, $PA = PB$, hence $PA^2 = PB^2$:

$(x+3)^2 + y^2 = (x-3)^2 + y^2.$ …