CHAPTER 3: COORDINATE GEOMETRY

1. Introduction to Coordinate Geometry

• To locate the position of an object or a point precisely in a plane, a single reference point is insufficient. We require two independent pieces of information or references.
• Real-life examples:
  • Finding a house usually requires a Street Number and a House Number.
  • Describing a seating plan requires a Column Number and a Row Number.
• This system of geometry was initially developed by the French philosopher and mathematician René Descartes. The system used for describing the position of a point in a plane is called the Cartesian system in his honour.

2. The Cartesian System

The system is based on two perpendicular number lines intersecting each other.

• Coordinate Axes:
  • The horizontal number line is called the x-axis (denoted as X'X).
  • The vertical number line is called the y-axis (denoted as Y'Y).
• The Origin: The point where the two axes intersect is called the origin and is denoted by the letter O.
• Direction:
  • OX and OY represent the positive directions (positive numbers).
  • OX' and OY' represent the negative directions (negative numbers).
• Quadrants: The axes divide the plane into four distinct parts called quadrants. They are numbered anticlockwise starting from the positive x-axis:
  • Quadrant I: Enclosed by positive x-axis and positive y-axis.
  • Quadrant II: Enclosed by negative x-axis and positive y-axis.
  • Quadrant III: Enclosed by negative x-axis and negative y-axis.
  • Quadrant IV: Enclosed by positive x-axis and negative y-axis.
• The entire plane is referred to as the Cartesian plane, the coordinate plane, or the xy-plane.

3. Coordinates of a Point

The position of a point is defined by its distances from the axes.

• Abscissa (x-coordinate): The perpendicular distance of a point from the y-axis (measured along the x-axis).
  (Positive along the positive direction of the x-axis and negative along the negative direction).
• Ordinate (y-coordinate): The perpendicular distance of a point from the x-axis (measured along the y-axis).
  (Positive along the positive direction of the y-axis and negative along the negative direction).
• Notation: The coordinates of a point are written in brackets as (x, y), where the x-coordinate comes first, followed by the y-coordinate.
• Uniqueness: The order matters. For example, (3, 4) is not the same position as (4, 3). Generally, if x ≠ y, then (x, y) ≠ (y, x).

4. Specific Points and Sign Conventions