Class 9 Mathematics Notes Chapter 3 (Coordinate Geometry) – Mathematics Book

Alright class, let's begin our detailed study of Chapter 3: Coordinate Geometry. This is a fundamental chapter that bridges algebra and geometry, providing a system to describe the position of points in a plane. Understanding this well is crucial, not just for Class 9, but for many concepts you'll encounter later, and it frequently appears in various government exams.

Chapter 3: Coordinate Geometry - Detailed Notes for Exam Preparation

1. Introduction: What and Why?

• Coordinate Geometry: A system used to locate the position of a point in a plane (or space) using numerical coordinates. It connects algebra and geometry.
• Need: To describe the exact location of an object or a point on a flat surface (a plane). Think about locating a seat in a theatre (Row number, Seat number) or a house on a map (Latitude, Longitude).
• Founder: Rene Descartes, a French mathematician, formalized this system, hence it's often called the Cartesian System.

2. The Cartesian Plane (or Coordinate Plane)

• It's a two-dimensional plane formed by the intersection of two perpendicular number lines.

• Axes:

  • X-axis: The horizontal number line. Positive numbers are to the right of the intersection point, negative numbers are to the left.
  • Y-axis: The vertical number line. Positive numbers are above the intersection point, negative numbers are below.

• Origin:

  • The point where the X-axis and Y-axis intersect.
  • Denoted by 'O'.
  • Its coordinates are (0, 0).

• Quadrants:

  • The two axes divide the plane into four regions called quadrants.
  • They are numbered I, II, III, and IV in an anti-clockwise direction, starting from the top right.
  • Quadrant I: Top right region. Both x and y coordinates are positive (+, +).
  • Quadrant II: Top left region. x is negative, y is positive (-, +).
  • Quadrant III: Bottom left region. Both x and y coordinates are negative (-, -).
  • Quadrant IV: Bottom right region. x is positive, y is negative (+, -).

  Quadrant	x-coordinate	y-coordinate	Example Point
  I	+ (Positive)	+ (Positive)	(3, 5)
  II	- (Negative)	+ (Positive)	(-2, 4)
  III	- (Negative)	- (Negative)	(-1, -3)
  IV	+ (Positive)	- (Negative)	(6, -2)

3. Coordinates of a Point

• The position of any point in the Cartesian plane is determined by an ordered pair of numbers (x, y).
• x-coordinate (Abscissa):
  • It is the perpendicular distance of the point from the Y-axis, measured along the X-axis.
  • It tells you how far to move left (negative) or right (positive) from the origin.
• y-coordinate (Ordinate):
  • It is the perpendicular distance of the point from the X-axis, measured along the Y-axis.
  • It tells you how far to move down (negative) or up (positive) from the origin.
• Convention: The x-coordinate is always written first, followed by the y-coordinate, enclosed in parentheses and separated by a comma: (x, y). The order matters! (3, 2) is different from (2, 3).

4. Points on the Axes

• Points on the X-axis: Any point lying directly on the X-axis has a y-coordinate of 0. Its coordinates are of the form (x, 0).
  • Example: (5, 0), (-3, 0).
  • The distance from the X-axis is zero.
• Points on the Y-axis: Any point lying directly on the Y-axis has an x-coordinate of 0. Its coordinates are of the form (0, y).
  • Example: (0, 4), (0, -2).
  • The distance from the Y-axis is zero.
• The Origin (0, 0): It lies on both the X-axis and the Y-axis.

5. Plotting a Point

• To plot a point P(x, y) on the Cartesian plane:
  1. Start at the Origin (0, 0).
  2. Move 'x' units along the X-axis (right if x is positive, left if x is negative).
  3. From that position, move 'y' units parallel to the Y-axis (up if y is positive, down if y is negative).
  4. Mark the final position as the point P(x, y).

Key Takeaways for Exams:

• Know the names: Cartesian Plane, X-axis, Y-axis, Origin, Quadrants, Abscissa, Ordinate.
• Remember the coordinates of the Origin: (0, 0).
• Memorize the signs of coordinates in each of the four quadrants (I: +,+; II: -,+; III: -,-; IV: +,-).
• Understand that the x-coordinate is the perpendicular distance from the Y-axis, and the y-coordinate is the perpendicular distance from the X-axis.
• Recognize the form of coordinates for points on the X-axis (x, 0) and Y-axis (0, y).
• Be precise with the order in the coordinate pair (x, y).

Multiple Choice Questions (MCQs)

Here are 10 MCQs to test your understanding. Try to solve them yourself before checking the answers.

1. The point where the X-axis and Y-axis intersect is called the:
   a) Abscissa
   b) Ordinate
   c) Origin
   d) Quadrant

2. What are the coordinates of the Origin in the Cartesian plane?
   a) (1, 1)
   b) (0, 0)
   c) (0, 1)
   d) (1, 0)

3. In which quadrant does the point (-4, 7) lie?
   a) Quadrant I
   b) Quadrant II
   c) Quadrant III
   d) Quadrant IV

4. The abscissa of the point (5, -3) is:
   a) 5
   b) -3
   c) 3
   d) -5

5. The ordinate of the point (-2, -6) is:
   a) -2
   b) 2
   c) 6
   d) -6

6. A point lies on the Y-axis at a distance of 4 units below the X-axis. What are its coordinates?
   a) (4, 0)
   b) (-4, 0)
   c) (0, 4)
   d) (0, -4)

7. The point (0, 9) lies on:
   a) The X-axis
   b) The Y-axis
   c) Quadrant I
   d) Quadrant IV

8. The signs of the abscissa and ordinate of a point in Quadrant III are:
   a) (+, +)
   b) (-, +)
   c) (-, -)
   d) (+, -)

9. The perpendicular distance of the point (7, 2) from the Y-axis is:
   a) 7 units
   b) 2 units
   c) 9 units
   d) 5 units

10. If the coordinates of a point are (0, -5), then it lies:
    a) In Quadrant II
    b) In Quadrant IV
    c) On the X-axis
    d) On the Y-axis

Answers to MCQs:

1. c) Origin
2. b) (0, 0)
3. b) Quadrant II (x is negative, y is positive)
4. a) 5 (The x-coordinate)
5. d) -6 (The y-coordinate)
6. d) (0, -4) (On Y-axis means x=0, below X-axis means y is negative)
7. b) The Y-axis (x-coordinate is 0)
8. c) (-, -)
9. a) 7 units (Perpendicular distance from Y-axis is the absolute value of the x-coordinate)
10. d) On the Y-axis (x-coordinate is 0)

Revise these concepts thoroughly. Coordinate geometry is visual, so practice plotting points and identifying their locations on a graph paper if you find it helpful. Good luck with your preparation!