CBSE Class 9 Maths Chapter 3 Coordinate Geometry Notes

Key Points to Remember

• Coordinate Geometry is a branch of mathematics that uses coordinates to locate points and define shapes on a plane.
• The Cartesian System uses two perpendicular lines (axes) to locate points.
• The horizontal line is the X-axis (or abscissa axis) and the vertical line is the Y-axis (or ordinate axis).
• The point where the X-axis and Y-axis intersect is called the Origin (0,0).
• A point in the plane is represented by an ordered pair (x, y), where 'x' is the x-coordinate and 'y' is the y-coordinate.
• The plane is divided into four quadrants by the axes: Quadrant I (+,+), Quadrant II (-,+), Quadrant III (-,-), and Quadrant IV (+,-).
• Points on the X-axis have their y-coordinate as 0 (e.g., (x, 0)). Points on the Y-axis have their x-coordinate as 0 (e.g., (0, y)).
• The distance of a point from the Y-axis is its x-coordinate, and its distance from the X-axis is its y-coordinate.

Essential Definitions

Cartesian System

A coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

Coordinate Axes

The two perpendicular number lines (X-axis and Y-axis) that form the Cartesian plane and are used to locate points.

Origin

The point of intersection of the X-axis and Y-axis, denoted by O(0,0).

Coordinates

An ordered pair (x, y) that specifies the location of a point in a Cartesian plane.

Abscissa

The x-coordinate of a point, representing its horizontal distance from the Y-axis.

Ordinate

The y-coordinate of a point, representing its vertical distance from the X-axis.

Quadrants

The four regions into which the Cartesian plane is divided by the coordinate axes.

Coordinate Plane

The plane containing the coordinate axes, also known as the Cartesian plane or XY-plane.

Understanding the Cartesian Plane and Plotting Points

The Cartesian plane, named after the French mathematician René Descartes, is a fundamental concept in coordinate geometry. It consists of two perpendicular number lines that intersect at their zeros. The horizontal line is called the X-axis (or x-axis), and the vertical line is called the Y-axis (or y-axis). Their point of intersection is known as the Origin, denoted by O and having coordinates (0,0). These two axes divide the plane into four regions, called quadrants.

In the Cartesian system, any point in the plane can be uniquely identified by an ordered pair of numbers (x, y). The first number, 'x', is called the abscissa and represents the point's perpendicular distance from the Y-axis. A positive 'x' means the point is to the right of the Y-axis, while a negative 'x' means it's to the left. The second number, 'y', is called the ordinate and represents the point's perpendicular distance from the X-axis. A positive 'y' means the point is above the X-axis, and a negative 'y' means it's below.

The signs of the coordinates define the quadrant a point lies in:

• Quadrant I: x > 0, y > 0 (e.g., (3, 2))
• Quadrant II: x < 0, y > 0 (e.g., (-4, 1))
• Quadrant III: x < 0, y < 0 (e.g., (-2, -5))
• Quadrant IV: x > 0, y < 0 (e.g., (5, -3))

Points lying on the X-axis have their ordinate (y-coordinate) as 0, for example, (5, 0). Points lying on the Y-axis have their abscissa (x-coordinate) as 0, for example, (0, -3). The origin (0,0) is the only point common to both axes. This system provides a powerful way to translate geometric problems into algebraic ones and vice versa.

Steps to Plot a Point (x, y) on a Cartesian Plane

1. —
2. —
3. —
4. —
5. —

Worked Examples

• Example 1: Plotting a Point Problem: Plot the point P(3, -2). Solution: Start at the origin. Move 3 units right along the X-axis (since x=3). From there, move 2 units down parallel to the Y-axis (since y=-2). Mark this final position as P(3, -2).
• Example 2: Identifying Quadrant Problem: In which quadrant does the point Q(-5, 4) lie? Solution: The x-coordinate is negative (-5) and the y-coordinate is positive (4). Points with (-, +) signs lie in Quadrant II.
• Example 3: Points on Axes Problem: Where do the points A(0, 5) and B(-3, 0) lie? Solution: For A(0, 5), the x-coordinate is 0, so it lies on the Y-axis. For B(-3, 0), the y-coordinate is 0, so it lies on the X-axis.

Exam Tip: Common Mistakes to Avoid

Students often confuse the order of coordinates, mistakenly plotting (y, x) instead of (x, y). Always remember that the abscissa (x-coordinate) comes first, followed by the ordinate (y-coordinate). Another common error is misinterpreting the signs for quadrants; double-check that you associate (+,+) with Q1, (-,+) with Q2, (-,-) with Q3, and (+,-) with Q4. Clearly label your axes and origin when drawing diagrams to avoid losing marks.

Quick Revision Checks

• Q: What are the coordinates of the origin? A: (0, 0)
• Q: If a point has coordinates (-2, -7), in which quadrant does it lie? A: Quadrant III
• Q: What is the perpendicular distance of the point (4, -3) from the Y-axis? A: 4 units (which is its x-coordinate)
• Q: A point has its x-coordinate as 0 and y-coordinate as -6. Where does this point lie? A: On the Y-axis (specifically, on the negative part of the Y-axis)

Frequently Asked Questions