Collinearity Slope Condition — From Intuition to Precision
Imagine three points scattered on a sheet of paper. You can always draw a triangle through them. But if those three points happen to lie perfectly on a single straight line — like beads on a thread — they are called collinear (from Latin co- meaning "together" and linearis meaning "belonging to a line").
The question is: how do you check, using only coordinates, whether three given points are collinear?
The Intuition
Think about walking from point A to point B, then from point B to point C. If all three lie on the same line, your direction of travel should not change when you turn at B. In other words, the slope of AB must equal the slope of BC.
Slope measures steepness: runrise=x2−x1y2−y1. If two segments share the same slope and meet at a common point (B), they lie on the same straight line.
This works because a line has a constant slope everywhere. If AB and BC have the same slope, they are parts of the same line — they cannot bend.
The Precise Statement
Let three points be A(x1,y1), B(x2,y2), and C(x3,y3). They are collinear if and only if:
x2−x1y2−y1=x3−x2y3−y2
provided that x1=x2 and x2=x3 (i.e., no vertical segment).
Collinearity Slope Condition
x2−x1y2−y1=x3−x2y3−y2
What About Vertical Lines?
If x1=x2, the slope of AB is undefined (division by zero). But the condition still works: if AB is vertical, then for collinearity, BC must also be vertical — meaning x2=x3. So the condition becomes: either both slopes are equal and defined, or both are undefined (i.e., both segments are vertical).
A Cleaner Algebraic Form
Cross-multiplying the slope equality gives a form that avoids division entirely:
(y2−y1)(x3−x2)=(y3−y2)(x2−x1)
This works for all cases, including vertical lines.
For quick checks, use the cross-multiplied form — no need to worry about zero denominators.
Example
Check if A(1,2), B(3,6), C(5,10) are collinear.
Slope of AB: 3−16−2=24=2
Slope of BC: 5−310−6=24=2
Slopes are equal → points are collinear. Indeed, they all lie on y=2x.
Why This Matters
The slope condition is the simplest test for collinearity in coordinate geometry. It appears in:
- Proving that three points lie on a line
- Checking if a point lies on a line through two other points
- Solving geometry problems in exams (Class 10, JEE, etc.)
A common mistake: checking only AB and BC slopes but forgetting that the points must be taken in order. If you check AB and AC, the condition still works — just ensure you use the same reference point. The safest pair is AB and BC (or AB and AC), always sharing one common point.
The Big Picture
Collinearity is about consistency of direction. The slope condition captures that: if the direction from A to B matches the direction from B to C, the three points cannot form a triangle — they are forced onto a single line.
The Collinearity Slope Condition is a direct application of the NCERT Class 11 Mathematics chapter on Straight Lines, matching searches like "condition for three points to be collinear" or "straight lines important questions class 11 maths". This equal-slope test is a common, quick-scoring proof-type question in CBSE boards and appears in JEE Main and CET coordinate geometry problems as well.