Locus and its Equation

A locus (plural: loci) is the set of all points that satisfy one or more given geometric conditions. Rather than describing a shape by listing individual points, a locus defines it by a rule — every point that obeys the rule belongs to the locus, and no other points do. For example, the set of all points exactly 5 units from a fixed centre forms a circle; the locus IS that circle. In coordinate geometry, we translate the geometric condition into an algebraic equation in x and y. This equation is called the equation of the locus. Every point (x, y) whose coordinates satisfy the equation lies on the locus, and every point on the locus satisfies the equation. This two-way correspondence is the heart of analytic geometry, uniting algebra and geometry. Finding the equation of a locus is a fundamental skill that underpins the study of conic sections (circles, parabolas, ellipses, hyperbolas) and many other curves. The method always follows the same logical pathway: name a general point, write the geometric condition algebraically, simplify, and interpret the resulting curve.

Key takeaways

Name a General Point
Write the Geometric Condition
Apply the Distance Formula or Relevant Formula
Simplify the Algebraic Expression
State and Verify the Locus

Name a General Point

Let P(x, y) represent any arbitrary point that might lie on the locus. Using a variable point rather than a specific one keeps the algebra general and produces an equation valid for all such points.

Write the Geometric Condition

Express the given condition in words first, then translate it into algebra. Common conditions include: a fixed distance from a point (use the distance formula), equal distances from two points (use distance formula twice and set them equal), a fixed ratio of distances (section formula or ratio of distance formulas), or a perpendicularity/collinearity condition (slope relationships).

Apply the Distance Formula or Relevant Formula

The distance between P(x, y) and a fixed point A(a, b) is PA = \sqrt{(x-a)^2 + (y-b)^2}. Substitute this into the condition from Step 2. If the condition involves two distances PA and PB, set up the equation PA = PB, PA = k·PB, or PA^2 + PB^2 = c, depending on the problem.

Simplify the Algebraic Expression

Square both sides to remove square roots (valid because distances are non-negative), expand brackets, collect like terms, and simplify to standard form. Check whether the result represents a recognisable curve: a linear equation gives a straight line, a second-degree equation with equal coefficients for x² and y² (and no xy term) gives a circle, and other second-degree equations give other conics.

State and Verify the Locus

Name the curve (e.g., 'the locus is a circle with centre (2, −3) and radius 5'). As a check, substitute one or two specific points that clearly satisfy the original geometric condition and confirm they also satisfy your equation. This verification step catches algebraic errors and builds geometric intuition.

The Key Equation

PA = PB \implies \sqrt{(x-x_1)^2+(y-y_1)^2} = \sqrt{(x-x_2)^2+(y-y_2)^2}

P(x,y) — General point on locus (none): An arbitrary point whose coordinates satisfy the locus condition
A(x_1, y_1) — Fixed point A (none): First given fixed point in the plane
B(x_2, y_2) — Fixed point B (none): Second given fixed point in the plane
PA — Distance from P to A (units): Euclidean distance calculated via the distance formula
PB — Distance from P to B (units): Euclidean distance calculated via the distance formula

Worked Example

Problem. Find the equation of the locus of a point P that moves so that its distance from A(1, 2) is always equal to its distance from B(5, −2). Identify the curve.

x - y - 3 = 0

Answer. The locus is the straight line x − y − 3 = 0, which is the perpendicular bisector of segment AB.

Figure

Diagram

Origins · 1 minRené Descartes and how this was discovered

René Descartes

1596–1650 · French

Descartes invented the Cartesian coordinate system (named after him) and published it in his 1637 work La Géométrie, an appendix to his Discours de la méthode. By assigning numerical coordinates to geometric points, he showed that any curve could be described by an algebraic equation — founding analytic geometry and making the study of loci systematic and algebraic for the first time. Descartes famously wrote much of his philosophy while lying in bed in the morning. Legend has it he observed a fly on his bedroom ceiling and realised its position could be described by two numbers — directly inspiring the coordinate system. While likely apocryphal, the story charmingly captures how simple observation can spark mathematical revolutions.

Distance Formula — The distance formula is the primary algebraic tool used in almost every locus problem to express a distance condition between a general point and fixed points.
Equation of a Circle — A circle is defined as the locus of points at a constant distance (radius) from a fixed centre — it is the most fundamental locus in coordinate geometry.
Equation of a Straight Line — Many loci (such as the perpendicular bisector) turn out to be straight lines, requiring fluency with linear equations in the Cartesian plane.
Conic Sections — Parabolas, ellipses, and hyperbolas are all defined as loci satisfying specific distance conditions; understanding locus is essential before studying conics.
Midpoint Formula — The midpoint formula is used in locus problems where a point traces a path related to the midpoint of a moving segment.