Coordinate Geometry Class 9 Notes Maths Chapter 7

Facts that Matter

Coordinate plane

• The position of a point on a plane is located using two mutually perpendicular lines.
• The plane formed by these two perpendicular lines is called the cartesian or coordinate plane and the lines are called axes.
• The horizontal axis is called the x-axis and the vertical axis is called the y-axis.
• The first number in an ordered pair is the x-coordinate, also called the abscissa.
• The second number in an ordered pair is the y-coordinate, also called the ordinate.
• In an ordered pair (x, y), the order is fixed; therefore, (3, 2) and (2, 3) represent two entirely different locations on the coordinate plane.
• Every point on the y-axis has abscissa zero. Every point on the x-axis has ordinate zero.
• The point where the axes meet is the origin; its coordinates are (0, 0).
• The axes divide the plane into four quadrants.
• Points with coordinates of the form 
  (+, +) lie in Quadrant I; 
  (-, +) in Quadrant II; 
  (-, -) in Quadrant III; 
  (+, -) in Quadrant IV.
• A point in the plane can be represented by an ordered pair of real numbers (x, y); this branch of mathematics is called Coordinate Geometry.

Cartesian System

• Two number lines drawn at right angles to each other are called axes.
• The horizontal line XOX' is the x-axis.
• The vertical line YOY' is the y-axis.
• Both axes lie in the same plane, called the cartesian plane, coordinate plane, or the XY-plane.

Let us note the following points:

• The perpendicular distance of a point from the y-axis is its x-coordinate (abscissa).
• The perpendicular distance of a point from the x-axis is its y-coordinate (ordinate).
• The abscissa of every point on the y-axis is zero.
• The ordinate of every point on the x-axis is zero.
• The axes intersect at the origin.
• The coordinates of the origin are (0, 0).

The positive direction of the x-axis is from the origin towards OX and the negative direction is towards OX'. The positive direction of the y-axis is from the origin towards OY and the negative direction is towards OY'.

Quadrants

The coordinate axes divide the plane into four regions called quadrants. They are numbered I, II, III and IV in anticlockwise order starting from the region where both x and y are positive.

• Quadrant I contains points with coordinates of the form (+, +).
• Quadrant II contains points with coordinates of the form (-, +).
• Quadrant III contains points with coordinates of the form (-, -).
• Quadrant IV contains points with coordinates of the form (+, -).

Plotting a Point in the Plane when Coordinates of the Point are Given

To plot a point whose coordinates are given as an ordered pair (x, y), follow these steps:

1. Draw the x-axis (horizontal) and the y-axis (vertical) so that they intersect at the origin O. Mark equal units on both axes in the positive and negative directions.
2. Starting at the origin, move along the x-axis to the x-coordinate value. Move right for positive x and left for negative x. Mark this position by dropping a perpendicular to the y-direction.
3. From that x-position, move parallel to the y-axis to reach the y-coordinate value. Move up for positive y and down for negative y. The point you reach is (x, y).

Example 1 - Plot the point (3, 2)

Draw axes and mark units. From the origin move 3 units to the right. From that position move 2 units up. Mark the point and label it P(3, 2).

Example 2 - Points on axes

• Any point on the x-axis has the form (a, 0) where a is the abscissa.
• Any point on the y-axis has the form (0, b) where b is the ordinate.

Quick Applications and Further Topics

• Coordinate Geometry helps solve geometric problems using algebra - locating points, finding distances between points, midpoints, slopes of lines, and equations of lines.
• The distance formula, midpoint formula, and gradient (slope) of a line are the next topics typically studied after plotting points and understanding coordinates.
• Understanding coordinates is useful for graphing functions, geometry constructions, navigation on maps, and problem solving in physics and engineering.

Summary: The cartesian plane uses two perpendicular axes to represent every point by an ordered pair (x, y). The x-value is the abscissa and the y-value is the ordinate. Knowing how to read and plot ordered pairs, identify quadrants, and understand axis behaviour is the foundation of coordinate geometry.