Class 9 Mathematics Notes Chapter 3 (Chapter 3) – Examplar Problem (Englisha) Book

Alright class, let's focus on Chapter 3: Coordinate Geometry from your NCERT Exemplar. This chapter is fundamental, and understanding it well is crucial not just for your class exams but also forms the base for concepts tested in various government examinations. We'll break down the key ideas precisely.

Chapter 3: Coordinate Geometry - Detailed Notes for Exam Preparation

1. Introduction: The Cartesian System

• Concept: A system used to locate the position of a point in a plane using two perpendicular lines. Developed by René Descartes, hence called the Cartesian system or Coordinate system.
• Components:
  • Axes: Two perpendicular number lines intersecting at their zeros.
    • X-axis: The horizontal number line.
    • Y-axis: The vertical number line.
  • Origin (O): The point where the X-axis and Y-axis intersect. Its coordinates are (0, 0).
• Plane: The flat surface on which the axes are drawn is called the Cartesian plane or the coordinate plane or the xy-plane.

2. Coordinates of a Point

• Concept: The position of any point in the Cartesian plane is determined by a pair of numbers called its coordinates. These are written as an ordered pair (x, y).
• Abscissa (x-coordinate): The perpendicular distance of the point from the Y-axis, measured along the X-axis. It is positive to the right of the Y-axis and negative to the left.
• Ordinate (y-coordinate): The perpendicular distance of the point from the X-axis, measured along the Y-axis. It is positive above the X-axis and negative below it.
• Convention: The abscissa (x-coordinate) is always written first, followed by the ordinate (y-coordinate), enclosed in parentheses, e.g., P(x, y).

3. Quadrants

• Concept: The two axes divide the Cartesian plane into four regions called quadrants.
• Numbering: They are numbered I, II, III, and IV in an anti-clockwise direction starting from the top-right region (where both x and y are positive).
• Signs of Coordinates in Quadrants: This is extremely important for quick identification:
  • Quadrant I: x > 0, y > 0. Coordinates are (+, +).
  • Quadrant II: x < 0, y > 0. Coordinates are (-, +).
  • Quadrant III: x < 0, y < 0. Coordinates are (-, -).
  • Quadrant IV: x > 0, y < 0. Coordinates are (+, -).

4. Points on the Axes

• Points on the X-axis: Any point lying on the X-axis has its perpendicular distance from the X-axis equal to zero. Therefore, its ordinate (y-coordinate) is always 0. The coordinates are of the form (x, 0).
• Points on the Y-axis: Any point lying on the Y-axis has its perpendicular distance from the Y-axis equal to zero. Therefore, its abscissa (x-coordinate) is always 0. The coordinates are of the form (0, y).
• Origin: The origin (0, 0) lies on both axes.

5. Plotting Points

• To plot a point P(x, y):
  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 on the X-axis, 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).

6. Distance from Axes

• The distance of a point P(x, y) from the Y-axis is the absolute value of its abscissa, i.e., |x|.
• The distance of a point P(x, y) from the X-axis is the absolute value of its ordinate, i.e., |y|.
  • Note: Distance is always non-negative.

7. Mirror Images (Reflection)

• This is a common concept tested:
  • The mirror image (reflection) of a point P(x, y) across the X-axis is the point P'(x, -y). (Ordinate sign changes).
  • The mirror image (reflection) of a point P(x, y) across the Y-axis is the point P''(-x, y). (Abscissa sign changes).
  • The mirror image (reflection) of a point P(x, y) across the Origin is the point P'''(-x, -y). (Both signs change).

Key Takeaways for Government Exams:

• Quickly identify the quadrant or axis of a point based on its coordinates.
• Understand the terms 'abscissa' and 'ordinate'.
• Know the coordinates of the origin and points on the axes.
• Be able to determine the distance of a point from the axes.
• Basic reflection concepts might be tested.
• Questions are usually direct and test fundamental understanding.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Chapter 3 concepts, similar to what you might encounter:

1. The point (-5, 3) lies in which quadrant?
   (a) I
   (b) II
   (c) III
   (d) IV

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. 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)

4. The abscissa of a point is its perpendicular distance from:
   (a) The Origin
   (b) The X-axis
   (c) The Y-axis
   (d) None of these

5. If the coordinates of a point are (0, -7), then the point lies:
   (a) In Quadrant IV
   (b) On the X-axis
   (c) On the Y-axis
   (d) In Quadrant III

6. What is the distance of the point P(3, -4) from the X-axis?
   (a) 3 units
   (b) -3 units
   (c) 4 units
   (d) -4 units

7. Signs of the abscissa and ordinate of a point in the IV quadrant are respectively:
   (a) +, +
   (b) -, +
   (c) +, -
   (d) -, -

8. What is the mirror image of the point (2, -5) in the Y-axis?
   (a) (-2, -5)
   (b) (2, 5)
   (c) (-2, 5)
   (d) (-5, 2)

9. If the ordinate of a point is 6 and its abscissa is -3, then the coordinates of the point are:
   (a) (6, -3)
   (b) (6, 3)
   (c) (-3, -6)
   (d) (-3, 6)

10. Points (1, -1), (2, -2), (4, -5), (-3, -4)
    (a) All lie in Quadrant IV
    (b) Do not lie in the same Quadrant
    (c) All lie in Quadrant III
    (d) All lie in Quadrant II

Answer Key for MCQs:

1. (b)
2. (b)
3. (d)
4. (c)
5. (c)
6. (c) (Distance is |y| = |-4| = 4)
7. (c)
8. (a)
9. (d)
10. (b) (Points are in Q IV, Q IV, Q IV, Q III respectively)

Study these notes thoroughly. Practice plotting points and identifying their locations. Pay close attention to the signs and the definitions of abscissa and ordinate. Good luck!