Class 9 Maths Chapter 1: Coordinate Geometry
1. Core Definitions & Cartesian System
- Coordinate Plane (or Cartesian/xy-Plane): The two-dimensional surface where we locate points. It is defined by two perpendicular lines intersecting each other.
- Coordinate Axes:
- x-axis: The horizontal line in the plane.
- y-axis: The vertical line in the plane.
- Origin (O): The point where the x-axis and y-axis intersect. Its coordinates are always (0, 0).
- Coordinates of a Point (x, y):
- x-coordinate (Abscissa): The perpendicular distance of a point from the y-axis, measured along the x-axis.
- y-coordinate (Ordinate): The perpendicular distance of a point from the x-axis, measured along the x-axis.
2. Rules for Points on the Axes
- Points on the x-axis: Any point lying on the x-axis has a y-coordinate of 0. Its coordinates are written in the form (x, 0).
- If x > 0, the point lies to the right of the origin.
- If x < 0, the point lies to the left of the origin.
- Points on the y-axis: Any point lying on the y-axis has an x-coordinate of 0. Its coordinates are written in the form (0, y).
- If y > 0, the point lies above the origin.
- If y < 0, the point lies below the origin.
3. Quadrants and Sign Conventions
The coordinate axes divide the Cartesian plane into four parts called quadrants, numbered counter-clockwise:
| Quadrant | Description | Sign of x | Sign of y | Coordinates Form |
|---|---|---|---|---|
| Quadrant I | Upper-right region | Positive (+) | Positive (+) | (+, +) |
| Quadrant II | Upper-left region | Negative (-) | Positive (+) | (-, +) |
| Quadrant III | Lower-left region | Negative (-) | Negative (-) | (-, -) |
| Quadrant IV | Lower-right region | Positive (+) | Negative (-) | (+, -) |
4. The Rule of Ordered Pairs
In coordinate geometry, the order of the numbers in (x, y) is crucial:
- If x ≠ y, then (x, y) ≠ (y, x).
- (x, y) = (y, x) if and only if x = y.
5. Formulas for Distance Between Two Points
Depending on how the points are oriented in the Cartesian plane, we use different distance calculations:
A. Distance along a Horizontal Line
If two points have the same y-coordinate, i.e., A(x₁, y) and B(x₂, y), the line segment joining them is parallel to the x-axis.
- Formula: Distance = |x₂ – x₁| (the absolute difference of their x-coordinates).
B. Distance along a Vertical Line
If two points have the same x-coordinate, i.e., A(x, y₁) and B(x, y₂), the line segment joining them is parallel to the y-axis.
- Formula: Distance = |y₂ – y₁| (the absolute difference of their y-coordinates).
C. General Distance Formula (Baudhāyana–Pythagoras Theorem)
For any two random points P(x₁, y₁) and Q(x₂, y₂) in the plane, the distance is calculated using the Pythagoras theorem:
Distance (d) = √[(x₂ – x₁)² + (y₂ – y₁)²]
Note: It does not matter if the differences (x₂ – x₁) or (y₂ – y₁) are negative, because squaring them always yields a positive value.