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Locus — Definition, Examples & Geometry Rules

Locus

A word for a set of points that forms a geometric figure or graph. For example, a circle can be defined as the locus of points that are all the same distance from a given point.

Key Formula

Locus={(x,y)∣condition involving x and y}\text{Locus} = \{ (x, y) \mid \text{condition involving } x \text{ and } y \}
Where:
  • (x,y)(x, y) = A general point in the coordinate plane
  • ∣\mid = Read as 'such that'; separates the point from the condition it must satisfy
  • condition\text{condition} = A geometric rule or equation that every point on the locus must satisfy

Worked Example

Problem: Find the equation of the locus of all points that are exactly 5 units from the point (2, 3).
Step 1: Identify the condition. Every point (x, y) on the locus must be exactly 5 units from the center (2, 3).
(x−2)2+(y−3)2=5\sqrt{(x - 2)^2 + (y - 3)^2} = 5
Step 2: Square both sides to eliminate the square root.
(x−2)2+(y−3)2=25(x - 2)^2 + (y - 3)^2 = 25
Step 3: Recognize the shape. This is the standard equation of a circle with center (2, 3) and radius 5.
Step 4: Verify with a test point. The point (7, 3) should lie on the locus since it is 5 units to the right of (2, 3).
(7−2)2+(3−3)2=25+0=25✓(7 - 2)^2 + (3 - 3)^2 = 25 + 0 = 25 \checkmark
Answer: The locus is the circle (x−2)2+(y−3)2=25(x - 2)^2 + (y - 3)^2 = 25.

Another Example

This example shows a locus that produces a straight line rather than a curve, demonstrating that the locus concept is not limited to circles. It also connects to the important geometric idea of perpendicular bisectors.

Problem: Find the equation of the locus of all points that are equidistant from the points A(1, 0) and B(5, 0).
Step 1: Set up the condition. A point (x, y) is on the locus if its distance to A equals its distance to B.
(x−1)2+y2=(x−5)2+y2\sqrt{(x - 1)^2 + y^2} = \sqrt{(x - 5)^2 + y^2}
Step 2: Square both sides to remove the square roots.
(x−1)2+y2=(x−5)2+y2(x - 1)^2 + y^2 = (x - 5)^2 + y^2
Step 3: The y2y^2 terms cancel. Expand the remaining squares.
x2−2x+1=x2−10x+25x^2 - 2x + 1 = x^2 - 10x + 25
Step 4: Simplify by cancelling x2x^2 from both sides and solving for xx.
−2x+1=−10x+25  ⟹  8x=24  ⟹  x=3-2x + 1 = -10x + 25 \implies 8x = 24 \implies x = 3
Step 5: The locus is the vertical line x=3x = 3. This is the perpendicular bisector of the segment from A(1, 0) to B(5, 0), which makes geometric sense since every point on a perpendicular bisector is equidistant from the segment's endpoints.
Answer: The locus is the vertical line x=3x = 3 (the perpendicular bisector of segment AB).

Frequently Asked Questions

What is the difference between a locus and a graph?
A graph is the visual representation of all solutions to an equation or inequality plotted on the coordinate plane. A locus is the set of points satisfying a geometric condition, which may or may not start as an equation. In practice, once you translate a locus condition into an equation, its graph and locus are the same set of points.
How do you find the equation of a locus?
Start by letting (x, y) be any point on the locus. Translate the geometric condition (such as a distance requirement) into an algebraic equation involving x and y. Then simplify the equation. The result is the equation of the locus. Common tools include the distance formula, midpoint formula, and slope formula.
What is the plural of locus?
The plural of locus is 'loci,' pronounced LOH-sigh. This Latin plural appears often in geometry when referring to multiple sets of points, each satisfying a different condition. The intersection of two loci gives points that satisfy both conditions simultaneously.

Locus vs. Graph of an Equation

LocusGraph of an Equation
DefinitionSet of all points satisfying a geometric conditionSet of all points (x, y) that make an equation true
Starting pointBegins with a geometric rule (e.g., 'equidistant from two points')Begins with an algebraic equation (e.g., y = 2x + 1)
ProcessTranslate condition into an equation, then plotPlot points that satisfy the given equation directly
ResultA geometric figure (line, circle, parabola, etc.)A curve or region in the coordinate plane
Typical useGeometry proofs and constructionsAlgebra and coordinate geometry

Why It Matters

Locus problems appear throughout geometry courses, standardized tests, and physics (e.g., the path of a projectile is a locus). Understanding the locus concept helps you see that every geometric shape — circles, parabolas, ellipses — can be defined by a single rule about distances or angles. This way of thinking bridges pure geometry and coordinate algebra, giving you a powerful tool for deriving equations from geometric descriptions.

Common Mistakes

Mistake: Forgetting that the locus includes ALL points satisfying the condition, not just a few sample points.
Correction: A locus is a complete set — every point meeting the condition belongs to it, and every point on the figure satisfies the condition. After finding the equation, verify that no extra or missing points exist by testing boundary cases.
Mistake: Squaring both sides of a distance equation without checking for extraneous solutions.
Correction: Squaring can introduce false solutions. After deriving your locus equation, substitute a point or two back into the original condition to confirm they genuinely satisfy it.

Related Terms

  • Set — A locus is a specific type of set of points
  • Point — The fundamental element that makes up a locus
  • Circle — Locus of points equidistant from a center
  • Graph of an Equation or Inequality — Visual representation closely related to a locus
  • Geometric Figure — A locus defines a geometric figure
  • Parabola — Locus of points equidistant from a focus and directrix
  • Ellipse — Locus where the sum of distances to two foci is constant
  • Distance Formula — Key tool for translating locus conditions into equations