Co-Ordinate Geometry : SGT
Coordinate Geometry
Cartesian Coordinate System
| Term | Meaning |
|---|---|
| Cartesian Plane | Plane formed by X-axis and Y-axis intersecting at right angles |
| Origin | Point of intersection (0,0) |
| Abscissa | X-coordinate of a point |
| Ordinate | Y-coordinate of a point |
| Quadrants | Plane divided into 4 regions by axes |
| Quadrant | Sign of (x, y) |
|---|---|
| I | (+, +) |
| II | (–, +) |
| III | (–, –) |
| IV | (+, –) |
Quadrant sign pattern: (+,+), (–,+), (–,–), (+,–)
Always write coordinates in order as (x, y) while plotting.
Distance Formula
| Case | Formula |
|---|---|
| General case | d = √[(x₂ − x₁)² + (y₂ − y₁)²] |
| Parallel to X-axis | d = |x₂ − x₁| |
| Parallel to Y-axis | d = |y₂ − y₁| |
Distance = √[(change in x)² + (change in y)²]
Always subtract carefully, squaring removes negative sign.
Section Formula
| Type | Formula |
|---|---|
| Internal division | ( (mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n) ) |
| External division | ( (mx₂ − nx₁)/(m−n), (my₂ − ny₁)/(m−n) ) |
| Midpoint | ( (x₁+x₂)/2 , (y₁+y₂)/2 ) |
| Centroid of triangle | ( (x₁+x₂+x₃)/3 , (y₁+y₂+y₃)/3 ) |
Centroid divides medians in the ratio 2:1.
Internal division uses plus sign, external division uses minus sign.
Area of Triangle in Coordinate Plane
| Formula | Condition |
|---|---|
| ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)| | Coordinates known |
| Condition | Meaning |
|---|---|
| Area = 0 | Points are collinear |
If area = 0 → Points lie on the same straight line.
Apply modulus to avoid negative area values.
Straight Lines and Slope
| Concept | Formula |
|---|---|
| Slope of line | m = (y₂ − y₁)/(x₂ − x₁) |
| Parallel lines | m₁ = m₂ |
| Perpendicular lines | m₁ × m₂ = -1 |
| Type of Slope | Meaning |
|---|---|
| Positive | Line rises from left to right |
| Negative | Line falls from left to right |
| Zero | Horizontal line |
| Undefined | Vertical line |
Parallel → Same slope | Perpendicular → Product = −1
Check sign carefully while finding slope difference.