Introduction
Coordinate Geometry is the study of geometry using the coordinate system. It allows representation of geometric shapes algebraically and solving problems using algebra and formulas.
Cartesian Coordinate System
The plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). The point where they meet is called the origin (0, 0).
Any point is represented as (x, y), where x is the abscissa and y is the ordinate.
Quadrants
- Quadrant I: (+, +) — Both x and y are positive.
- Quadrant II: (−, +) — x is negative, y is positive.
- Quadrant III: (−, −) — Both x and y are negative.
- Quadrant IV: (+, −) — x is positive, y is negative.
Distance Formula
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Example: Distance between P(3, 4) and Q(7, 1) = √[(4)² + (−3)²] = √[16 + 9] = √25 = 5 units.
Section Formula
P(x, y) = ( (mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n) )
Example: A(2, 3), B(4, 7), ratio 1:3 → P = (2.5, 4)
Midpoint Formula
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
Example: Midpoint of (6, 2) and (4, 8) is (5, 5)
Slope of a Line
m = (y₂ − y₁) / (x₂ − x₁)
Example: Slope of line joining (2,3) and (5,11) = (11−3)/(5−2) = 8/3
Equation of a Line
y − y₁ = m(x − x₁)
This is the point-slope form of a line’s equation.
Collinearity of Points
Three points are collinear if the area of triangle formed by them is zero.
Area = 1/2 |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Applications
Coordinate Geometry is used in navigation, computer graphics, architecture, engineering design, robotics, and physics simulations.
Practice Questions
- Find the distance between (−3, 2) and (4, −5).
- Find the coordinates of the midpoint between (1, −1) and (3, 5).
- Find the slope of the line joining points (4, 7) and (6, 15).
- If P divides AB in the ratio 2:5 where A(1, 2) and B(9, 5), find P.
- Show that points (1, 2), (3, 6), (5, 10) are collinear.