CBSE Class 9 Maths Notes: Coordinate Geometry

Definition of the Cartesian Plane

The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane formed by the intersection of two perpendicular number lines. This system provides a way to represent points using ordered pairs, making it easier to analyze geometric shapes and relationships algebraically. It’s a fundamental concept in coordinate geometry, allowing us to visualize and solve mathematical problems.

Axes, Origin, and Quadrants

The two perpendicular number lines are called the axes.

• The horizontal axis is called the x-axis (also known as the abscissa).
• The vertical axis is called the y-axis (also known as the ordinate).

The point where the x-axis and y-axis intersect is called the origin, and it has coordinates (0, 0). The axes divide the plane into four regions called quadrants:

• Quadrant I: x > 0, y > 0
• Quadrant II: x < 0, y > 0
• Quadrant III: x < 0, y < 0
• Quadrant IV: x > 0, y < 0

Ordered Pairs and Plotting Points

An ordered pair (x, y) represents a point on the Cartesian plane. The first number, ‘x’, is the x-coordinate (or abscissa), and the second number, ‘y’, is the y-coordinate (or ordinate).

Plotting a point involves the following steps:

1. Start at the origin (0, 0).
2. Move horizontally along the x-axis: right for positive x-values, left for negative x-values.
3. Move vertically along the y-axis: up for positive y-values, down for negative y-values.
4. Mark the point where you end up.

Example: To plot the point (3, -2), move 3 units to the right along the x-axis and 2 units down along the y-axis.

Abscissa and Ordinate

As mentioned earlier, in an ordered pair (x, y):

• The abscissa is the x-coordinate, representing the horizontal distance from the y-axis.
• The ordinate is the y-coordinate, representing the vertical distance from the x-axis.

For example, in the point (5, -7), the abscissa is 5, and the ordinate is -7.

Core Principles and Formulas

The core principle is representing points using ordered pairs and understanding their relationship within the Cartesian plane. The main concepts involved are based around:

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the Cartesian plane is given by: $d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
• Section Formula: Used to find the coordinates of a point that divides a line segment. For a point dividing the line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m:n$ is: $(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n})$
• Midpoint Formula: The midpoint of the line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is: $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$