Coordinate Geometry

Explore the mesmerizing world of straight lines through interactive animations

Straight Lines in Motion

Discover how changing parameters affects the graphical representation of lines in the Cartesian plane. Play with the controls below to see real-time changes!

Dancing Line Equations

Each form of line equation has its own unique properties and applications. Click through the tabs to explore them all!

Slope-Intercept Form

y = mx + b

Where:

  • m is the slope of the line (steepness and direction)
  • b is the y-intercept (where the line crosses the y-axis)

Example

Find the equation of a line with slope 2 and y-intercept -3.

Solution: y = 2x - 3

Point-Slope Form

y - y₁ = m(x - x₁)

Where:

  • m is the slope of the line
  • (x₁, y₁) is a known point on the line

Example

Find the equation of a line with slope -1 passing through point (2, 3).

Solution: y - 3 = -1(x - 2) → y = -x + 5

Two-Point Form

(y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)

Where:

  • (x₁, y₁) and (x₂, y₂) are two known points on the line

Example

Find the equation of a line passing through points (1, 2) and (3, 4).

Solution: (y - 2)/(4 - 2) = (x - 1)/(3 - 1) → y = x + 1

Intercept Form

x/a + y/b = 1

Where:

  • a is the x-intercept (where the line crosses the x-axis)
  • b is the y-intercept (where the line crosses the y-axis)

Example

Find the equation of a line with x-intercept 3 and y-intercept 4.

Solution: x/3 + y/4 = 1 → 4x + 3y = 12

Normal Form

x cos θ + y sin θ = p

Where:

  • θ is the angle the normal makes with the positive x-axis
  • p is the length of the perpendicular from the origin to the line

Example

Find the equation of a line where the perpendicular from origin is 5 units and makes 30° with x-axis.

Solution: x cos30° + y sin30° = 5 → (√3/2)x + (1/2)y = 5

Interactive Concepts

Angle Between Two Lines

tan θ = |(m₁ - m₂)/(1 + m₁m₂)|

Calculate the angle between two lines given their slopes m₁ and m₂.

Distance from Point to Line

d = |Ax₁ + By₁ + C|/√(A² + B²)

Find the shortest distance from point (x₁, y₁) to line Ax + By + C = 0.

Family of Lines

L₁ + kL₂ = 0

Represents all lines passing through the intersection of L₁ and L₂, where k is a parameter.

Challenge Yourself

Interactive Practice