Line Equations in Coordinate Geometry
General Form: Ax + By + C = 0Slope-Intercept: y = mx + bPoint-Slope: y - y₁ = m(x - x₁)
📐General Form of a Line
Ax + By + C = 0
where A, B, and C are real constants, and A and B are not both zero.
A ≠ 0, B = 0
Vertical Line
x = -C/A
A = 0, B ≠ 0
Horizontal Line
y = -C/B
A ≠ 0, B ≠ 0
Oblique Line
slope = -A/B
🔍Key Properties and Formulas
Slope and Intercepts
Slope: m = -A/B (when B ≠ 0)
X-intercept: x = -C/A (when A ≠ 0)
Y-intercept: y = -C/B (when B ≠ 0)
Angle with x-axis: θ = arctan(-A/B)
Distance and Normal
Distance from origin:
d = |C|/√(A² + B²)
Normal vector: (A, B)
Direction vector: (-B, A)
📏Distance from Point to Line
d = |Ax₀ + By₀ + C|/√(A² + B²)
Distance from point (x₀, y₀) to line Ax + By + C = 0
When to use:
- • Finding shortest distance to a line
- • Checking if points are equidistant from a line
- • Determining position relative to a line
Sign interpretation:
- • Positive: Point on one side
- • Negative: Point on other side
- • Zero: Point on the line
⟂Parallel and Perpendicular Lines
Parallel Lines
A₁x + B₁y + C₁ = 0
A₂x + B₂y + C₂ = 0
Condition for parallel:
A₁/A₂ = B₁/B₂ ≠ C₁/C₂
Same slope, different intercepts
Perpendicular Lines
A₁x + B₁y + C₁ = 0
A₂x + B₂y + C₂ = 0
Condition for perpendicular:
A₁A₂ + B₁B₂ = 0
Product of slopes = -1
📝Different Forms of Line Equations
Slope-Intercept Form
y = mx + b
- • m = slope
- • b = y-intercept
- • Most common form
Point-Slope Form
y - y₁ = m(x - x₁)
- • (x₁, y₁) = known point
- • m = slope
- • Useful for construction
Two-Point Form
(y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)
- • Two points: (x₁, y₁), (x₂, y₂)
- • Direct from coordinates
🔬 Real-World Applications
- • Engineering: Structural analysis and design
- • Physics: Motion in straight lines
- • Economics: Linear regression and trends
- • Computer Graphics: Line rendering
🧮 Problem-Solving Tips
- • Always check if A and B are both zero
- • Use general form for easier calculations
- • Remember: slope = -A/B (when B ≠ 0)
- • Distance formula works for any point-line pair