Straight Lines

A straight line is the simplest curve in coordinate geometry. In Class 11, we study lines in the Cartesian plane, learning to write their equations in various forms and to find distances and angles between them. These concepts underpin analytic geometry and are fundamental for higher mathematics.

Slope of a Line:
The slope (or gradient) of a line measures its steepness. If a line passes through two points (x₁, y₁) and (x₂, y₂), then:
slope m = (y₂ - y₁) / (x₂ - x₁)

A horizontal line has slope 0.
A vertical line has undefined slope.
If two lines are parallel, their slopes are equal (m₁ = m₂).
If two lines are perpendicular, m₁ × m₂ = -1.

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Equations of a Line

Key formulas

Slope-point form: y - y₁ = m(x - x₁)

Slope-intercept form: y = mx + c, where c is the y-intercept

Two-point form: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)

Intercept form: x/a + y/b = 1, where a = x-intercept, b = y-intercept

Normal form: x cos(w) + y sin(w) = p, where p is the perpendicular distance from origin and w is the angle the perpendicular makes with x-axis

General form: Ax + By + C = 0

Distance of a Point from a Line:
The perpendicular distance of point (x₁, y₁) from line Ax + By + C = 0 is:
d = |Ax₁ + By₁ + C| / √(A² + B²)

Angle Between Two Lines:
tan(theta) = |(m₁ - m₂) / (1 + m₁ × m₂)|

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Worked Examples

Example 1

Find the slope of the line joining (2, 3) and (5, 9).
m = (9 - 3)/(5 - 2) = 6/3 = 2

Example 2

Write the equation of the line with slope 3 passing through (1, -2).
y - (-2) = 3(x - 1) => y + 2 = 3x - 3 => 3x - y - 5 = 0

Example 3

Find the equation of the line with x-intercept 4 and y-intercept -3.
x/4 + y/(-3) = 1 => 3x - 4y = 12 => 3x - 4y - 12 = 0

Example 4

Are the lines 2x + 3y - 5 = 0 and 4x + 6y + 1 = 0 parallel?
Slopes: m₁ = -2/3, m₂ = -4/6 = -2/3. Yes, since m₁ = m₂, the lines are parallel.

Example 5

Find the distance of point (3, -5) from the line 3x - 4y + 1 = 0.
d = |3(3) - 4(-5) + 1| / √(9 + 16) = |9 + 20 + 1| / 5 = 30/5 = 6

Example 6

Find the equation of the line perpendicular to 2x - 3y + 1 = 0 and passing through (4, 5).
Slope of given line = 2/3. Perpendicular slope = -3/2.
y - 5 = -3/2 × (x - 4) => 2y - 10 = -3x + 12 => 3x + 2y - 22 = 0

Example 7

Find the angle between lines y = 2x + 1 and y = x + 3.
m₁ = 2, m₂ = 1. tan(theta) = |(2 - 1)/(1 + 2)| = 1/3. So theta = arctan(1/3).

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Common mistakes

The slope formula requires (y₂ - y₁)/(x₂ - x₁), not the reverse. Be consistent with which point is (x₁, y₁).
When two lines are perpendicular, use m₁ × m₂ = -1, NOT m₁ × m₂ = 1.
The distance formula requires the absolute value of the numerator — never omit it.

Summary

Learn all six standard forms of the equation of a line. The distance formula and perpendicularity condition are the most commonly tested in board exams. Practice converting between forms.

Equations of a Line

Worked Examples

Practice Problems