Co-Ordinate Geometry
Coordinate Geometry
Introduction to Coordinate Geometry – Cartesian System
| Element | Description |
|---|---|
| Origin | Point where X-axis and Y-axis intersect (0,0) |
| X-axis | Horizontal number line |
| Y-axis | Vertical number line |
| Quadrants | Four regions formed by X and Y axes |
Memory Tip: Quadrants move anticlockwise: I (+,+), II (−,+), III (−,−), IV (+,−).
Exam Tip: Always check the signs of coordinates before identifying the quadrant.
Distance Between Two Points on Axes & Distance Formula
| Case | Formula |
|---|---|
| On X-axis | |x₂ − x₁| |
| On Y-axis | |y₂ − y₁| |
| General Case | √[(x₂ − x₁)² + (y₂ − y₁)²] |
Memory Tip: Distance formula is based on Pythagoras theorem.
Exam Tip: Always square inside first, then add, then take square root.
Section Formula & Centroid
| Concept | Formula |
|---|---|
| Internal Section | ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)) |
| External Section | ((mx₂ − nx₁)/(m−n), (my₂ − ny₁)/(m−n)) |
| Centroid of Triangle | ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) |
Memory Tip: Centroid divides median in the ratio 2:1 from vertex.
Exam Tip: Centroid is mean of all x’s and all y’s.
Trisection Points of a Line Segment
| Point | Ratio | Formula |
|---|---|---|
| First Trisection | 1:2 | ((1x₂+2x₁)/3 , (1y₂+2y₁)/3) |
| Second Trisection | 2:1 | ((2x₂+1x₁)/3 , (2y₂+1y₁)/3) |
Memory Tip: Trisection means dividing line into three equal parts.
Exam Tip: Use section formula with ratios 1:2 and 2:1.
Area of Triangle, Collinearity & Heron’s Formula
| Concept | Formula |
|---|---|
| Area using coordinates | ½ |x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)| |
| Collinearity condition | Area = 0 |
| Heron’s Formula | √[s(s−a)(s−b)(s−c)] |
Memory Tip: If area = 0, all three points lie on one straight line.
Exam Tip: Heron’s formula is used when coordinates first give side lengths.
Straight Line & Slope Concepts
| Concept | Formula |
|---|---|
| Slope | m = (y₂ − y₁)/(x₂ − x₁) |
| Equation of line | y − y₁ = m(x − x₁) |
| Horizontal line | Slope = 0 |
| Vertical line | Slope undefined |
Memory Tip: Horizontal → Zero slope, Vertical → Infinite / Undefined slope.
Exam Tip: If x₂ = x₁ → vertical line, if y₂ = y₁ → horizontal line.
Slope of Line Joining Two Points
| Points | Condition |
|---|---|
| Same x-coordinates | Line parallel to Y-axis |
| Same y-coordinates | Line parallel to X-axis |
| m₁ × m₂ = −1 | Lines are perpendicular |
Memory Tip: Negative reciprocal slopes indicate perpendicular lines.
Exam Tip: Always simplify the slope fraction before comparison.