Equation of a Line
Introduction
The equation of a line in coordinate geometry expresses the relationship between the x and y coordinates of every point on the line. In ICSE Class 10, you study various forms of the equation of a line and learn to determine conditions for lines to be parallel or perpendicular.
Slope (Gradient) of a Line
The slope of a line is the measure of its steepness. It is denoted by m.
m = (y₂ − y₁) / (x₂ − x₁), where (x₁, y₁) and (x₂, y₂) are two points on the line.
- A horizontal line has slope 0.
- A vertical line has undefined slope.
- Slope is positive when the line rises from left to right.
- Slope is negative when the line falls from left to right.
Forms of the Equation of a Line
1. Slope-Intercept Form
y = mx + c
Where m = slope, c = y-intercept (the point where the line crosses the y-axis).
2. Point-Slope Form
y − y₁ = m(x − x₁)
Where m = slope and (x₁, y₁) is a known point on the line.
3. Two-Point Form
(y − y₁) / (x − x₁) = (y₂ − y₁) / (x₂ − x₁)
Where (x₁, y₁) and (x₂, y₂) are two points on the line.
4. Intercept Form
x / a + y / b = 1
Where a = x-intercept and b = y-intercept.
5. General Form
Ax + By + C = 0
Where A, B, C are constants, and A and B are not both zero.
Conditions for Parallelism
Two lines are parallel if their slopes are equal.
If L₁: y = m₁x + c₁ and L₂: y = m₂x + c₂ Then L₁ ∥ L₂ ⇒ m₁ = m₂
Example: y = 3x + 2 is parallel to y = 3x − 5.
Conditions for Perpendicularity
Two lines are perpendicular if the product of their slopes is −1.
L₁ ⟂ L₂ ⇒ m₁ × m₂ = −1
This means m₂ = −1/m₁ (the slopes are negative reciprocals).
Example: y = 2x + 3 is perpendicular to y = −¹/₂x + 1.
Worked Examples
Example 1: Finding Equation Given Slope and Point
Find the equation of a line with slope −2 passing through (3, 5).
Solution: Using point-slope form: y − y₁ = m(x − x₁) y − 5 = −2(x − 3) y − 5 = −2x + 6 y = −2x + 11
Equation: y = −2x + 11 or 2x + y − 11 = 0
Example 2: Equation Through Two Points
Find the equation of the line passing through (2, 3) and (4, 7).
Solution: Slope m = (7 − 3) / (4 − 2) = 4 / 2 = 2
Using point-slope form with (2, 3): y − 3 = 2(x − 2) y − 3 = 2x − 4 y = 2x − 1
Equation: y = 2x − 1
Example 3: Parallel and Perpendicular Lines
Find the equation of a line through (1, −2) that is (a) parallel to y = 3x + 4, (b) perpendicular to y = 3x + 4.
Solution: (a) Parallel: slope m = 3 y − (−2) = 3(x − 1) y + 2 = 3x − 3 y = 3x − 5
(b) Perpendicular: slope m = −1/3 y − (−2) = (−1/3)(x − 1) y + 2 = (−1/3)x + 1/3 y = (−1/3)x − 5/3 or 3y + x + 5 = 0
Example 4: Intercept Form
A line cuts intercepts 3 and −4 on the axes. Find its equation.
Solution: Using intercept form: x/a + y/b = 1 x/3 + y/(−4) = 1 x/3 − y/4 = 1
Multiply by 12: 4x − 3y = 12
Equation: 4x − 3y = 12
Example 5: Finding Slope from General Form
Find the slope and y-intercept of the line 3x + 2y − 6 = 0.
Solution: Convert to slope-intercept form: 2y = −3x + 6 y = (−3/2)x + 3
Slope m = −3/2, y-intercept c = 3
Summary Table
| Form | Equation | When to Use |
|---|---|---|
| Slope-intercept | y = mx + c | When slope and y-intercept are known |
| Point-slope | y − y₁ = m(x − x₁) | When slope and one point are known |
| Two-point | (y − y₁)/(x − x₁) = (y₂ − y₁)/(x₂ − x₁) | When two points are known |
| Intercept | x/a + y/b = 1 | When both intercepts are known |
| General | Ax + By + C = 0 | Standard form for all lines |
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| Confusing slope formula order | m = (y₂ − y₁)/(x₂ − x₁) — y difference over x difference |
| Using wrong perpendicular slope condition | m₁ × m₂ = −1, not m₁ = m₂ |
| Forgetting to simplify the equation | Express in y = mx + c or Ax + By + C = 0 |
| Mixing x-intercept and y-intercept | x-intercept: y = 0; y-intercept: x = 0 |
ICSE Exam Focus
Equation of a line carries 6–8 marks in ICSE exams. Key question types:
- Finding equation from given conditions.
- Parallel and perpendicular lines.
- Finding slope and intercepts.
- Proving collinearity of points using slopes.
Marks Blueprint:
| Topic | Marks |
|---|---|
| Finding equation (slope + point) | 2 |
| Parallel/perpendicular lines | 3 |
| Intercept form problems | 2 |
| Conversion between forms | 2 |
Self-Test Questions
-
Find the equation of a line with slope 4 and passing through (−2, 3).
-
Find the equation of the line passing through (3, −1) and (5, 3).
-
Show that the lines 2x + 3y − 5 = 0 and 6x + 9y + 7 = 0 are parallel.
-
Find the equation of a line perpendicular to 3x − 4y + 12 = 0 and passing through (2, 5).
-
A line has x-intercept 4 and y-intercept −3. Find its equation.
-
Find the slope and y-intercept of the line 5x − 2y + 10 = 0.
In ICSE, the slope formula and conditions for parallelism/perpendicularity are fundamental. Memorise them thoroughly.
