In analytical geometry, a straight line represents the shortest distance between two points. Its equation helps us understand the position, slope, and behavior of the line in the coordinate plane.
Inclination of a Line
The inclination of a line is the angle θ made by the line with the positive X-axis, measured anticlockwise.
– If the line rises from left to right → θ is acute.
– If the line falls from left to right → θ is obtuse.
Slope of a Line
The slope (m) represents the steepness of the line.
m = tan θ
Slope Using Two Points
For points P(x₁, y₁) and Q(x₂, y₂):
m = (y₂ − y₁) / (x₂ − x₁)
Different Forms of Equation of a Straight Line
1. Slope–Intercept Form
y = mx + c
m = slope, c = y-intercept
2. Point–Slope Form
If line passes through (x₁, y₁) with slope m:
y − y₁ = m(x − x₁)
3. Two-Point Form
If the line passes through points (x₁, y₁) and (x₂, y₂):
(y − y₁) / (y₂ − y₁) = (x − x₁) / (x₂ − x₁)
4. Intercept Form
x/a + y/b = 1
a = x-intercept, b = y-intercept
5. Normal Form
x cos α + y sin α = p
p = perpendicular distance from origin, α = angle that perpendicular makes with X-axis.
General Form of a Straight Line
Ax + By + C = 0
Here A, B, C are real numbers and A² + B² ≠ 0.
Angle Between Two Lines
If slopes are m₁ and m₂:
tan θ = |(m₂ − m₁) / (1 + m₁m₂)|
Special Cases
- Lines are parallel if m₁ = m₂
- Lines are perpendicular if m₁m₂ = −1
Distance Formulas
1. Distance Between Two Points
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
2. Distance of a Point from a Line
For point (x₁, y₁) from line Ax + By + C = 0:
d = |Ax₁ + By₁ + C| / √(A² + B²)
3. Distance Between Two Parallel Lines
For lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0:
d = |C₂ − C₁| / √(A² + B²)
Condition for Collinearity of Three Points
Points (x₁, y₁), (x₂, y₂), (x₃, y₃) are collinear if:
(y₂ − y₁)/(x₂ − x₁) = (y₃ − y₂)/(x₃ − x₂)
Equation of a Line Parallel or Perpendicular to Another Line
Parallel Line
For reference line Ax + By + C = 0, a parallel line will have the form:
Ax + By + D = 0
Perpendicular Line
If original line is Ax + By + C = 0, then perpendicular line is:
Bx − Ay + K = 0
Angle of Intersection of Two Lines
When two lines intersect, the acute angle between them is:
tan θ = |(m₂ − m₁)/(1 + m₁m₂)|
Examples (Conceptual Overview)
- Finding the slope using two points
- Writing equation of line in slope–intercept form
- Finding distance of point from a line
- Determining if two lines are perpendicular or parallel
Detailed Notes:
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