Co-ordinate Geometry Chapter Notes | Mathematics Class 9 ICSE PDF Download

Introduction

Co-ordinate Geometry is a fascinating branch of mathematics where numbers and shapes come together to create a map of points in space. Imagine plotting a treasure map where each location is marked by a pair of numbers, guiding you through a grid of lines. This chapter introduces you to the art of locating points using coordinates and understanding their relationships with various geometric figures, all based on a system of perpendicular lines called axes. By mastering these concepts, you'll unlock the ability to visualize and solve problems involving positions and patterns in a plane, making math both logical and exciting!

• Co-ordinate Geometry uses two numbers, called coordinates, to pinpoint a location in a plane.
• It involves two perpendicular number lines, known as coordinate axes, to define positions.
• The study includes:
  • Locating points using coordinates (co-ordinate aspect).
  • Understanding how points relate to geometric shapes (geometry aspect).
• Example: A point at (3, 4) is located by moving 3 units right along the horizontal axis and 4 units up along the vertical axis, showing its position relative to the axes.

Dependent and Independent Variables

• In linear equations like 3x + 4y = 5, x and y are variables representing coordinates.
• Dependent variable: The variable whose value depends on the other (solved as the subject of the equation).
• Independent variable: The variable that determines the value of the dependent variable.
• Steps to identify variables:
  • If the equation is solved for y (e.g., y = mx + c), y is dependent, x is independent.
  • If solved for x (e.g., x = ay + b), x is dependent, y is independent.
• Example: For the equation 4x - 5y + 20 = 0: 
• (i) Solve for x: 4x = 5y - 20 → x = (5/4)y - 5. Here, x is dependent, y is independent.
• (ii) Solve for y: -5y = -4x - 20 → y = (4/5)x + 4. Here, y is dependent, x is independent.

Ordered Pair

An ordered pair is a set of two numbers written in a specific order, denoted as (a, b).
Steps to understand ordered pairs:

• Write numbers in a specific order, separated by a comma, inside parentheses.
• First number (a) is the first component, second number (b) is the second component.
• Order matters: (a, b) ≠ (b, a) unless a = b.
• Equal ordered pairs have equal corresponding components: (a, b) = (c, d) means a = c and b = d.
• Components can be equal, e.g., (5, 5).

Example: For (x, 4) = (-7, y), compare components: x = -7 and y = 4.

Cartesian Plane

• A Cartesian plane is formed by two perpendicular number lines intersecting at their zero points.
• Components:
  • Horizontal line (XOX'): x-axis.
  • Vertical line (YOY'): y-axis.
  • Intersection point (O): origin, with coordinates (0, 0).
• The x-axis and y-axis together form the Cartesian coordinate system.

Co-ordinates of Points

Each point in a Cartesian plane is represented by an ordered pair (x, y).
Steps to find coordinates:

• Measure the distance from the origin along the x-axis: this is the x-coordinate (abscissa).
• Measure the distance from the origin along the y-axis: this is the y-coordinate (ordinate).
• Write as (abscissa, ordinate), e.g., (x, y).

Example: For a point with abscissa 4 and ordinate 2, the coordinates are (4, 2).

Quadrants and Sign Convention

The coordinate axes divide the plane into four quadrants.

Quadrant naming (anti-clockwise from OX):

• First quadrant (XOY): Both x and y are positive.
• Second quadrant (X'OY): x is negative, y is positive.
• Third quadrant (X'OY'): Both x and y are negative.
• Fourth quadrant (Y'OX): x is positive, y is negative.

Sign convention:

• First quadrant: (+, +)
• Second quadrant: (-, +)
• Third quadrant: (-, -)
• Fourth quadrant: (+, -)

Example: Point A(4, 2) is in the first quadrant (x = 4, y = 2, both positive), while point C(-4, -5) is in the third quadrant (x = -4, y = -5, both negative).

Plotting of Points

• Plotting a point involves locating its position on the Cartesian plane using its coordinates.
• Steps to plot a point:
  • Start at the origin (0, 0).
  • Move along the x-axis (right for positive, left for negative) by the x-coordinate.
  • From there, move along the y-axis (up for positive, down for negative) by the y-coordinate.
  • Mark the point and label it.
• Special cases:
  • Origin: (0, 0).
  • Points on x-axis: (x, 0), e.g., (7, 0).
  • Points on y-axis: (0, y), e.g., (0, 8).

Example: To plot A(4, 2): 

• Move 4 units right along the x-axis.
• Move 2 units up parallel to the y-axis.
• Mark and label the point as A(4, 2).

Graphs of x=0, y=0, x=a, y=a, etc.

Equations of specific lines:

• x = 0: Represents the y-axis, as all points have x-coordinate 0 (e.g., (0, 7)).
• x = a: A line parallel to the y-axis at a distance of 'a' units (e.g., x = -5 for a line 5 units left of the y-axis).
• y = 0: Represents the x-axis, as all points have y-coordinate 0 (e.g., (8, 0)).
• y = a: A line parallel to the x-axis at a distance of 'a' units (e.g., y = 6 for a line 6 units above the x-axis).

Example: For y = 3, y + 5 =0, x = 4, x + 6= 0. The graphs are straight lines parallel to the x-axis at a distance of 3 units above it.

Graphing a Linear Equation

• A linear equation produces a straight-line graph.
• Steps to graph a linear equation:
  • Find at least three points that satisfy the equation by choosing values for x and calculating y.
  • Create a table of x and y values.
  • Plot the points on a graph and draw a straight line through them.
• Types of linear equations:
  • Type 1: y = mx (passes through the origin).
  • Type 2: y = mx + c (c is the y-intercept, not zero).
  • General form: ax + by + c = 0.
• Example: For y = -2x: 
• Choose x = 0: y = -2 × 0 = 0 → (0, 0).
• Choose x = 3: y = -2 × 3 = -6 → (3, -6).
• Choose x = -2: y = -2 × -2 = 4 → (-2, 4).
• Plot points (0, 0), (3, -6), (-2, 4) and draw a straight line through them.

Inclination and Slope

• Inclination: The angle (θ) a line makes with the positive x-axis, measured anti-clockwise.
• Slope (m): Defined as m = tan θ, where θ is the inclination.
• Special cases:
  • For x-axis or lines parallel to it: θ = 0°, so m = tan 0° = 0.
  • For y-axis or lines parallel to it: θ = 90°, so m = tan 90° = undefined.
• Example: If a line’s inclination is 45°, then θ = 45°, and slope m = tan 45° = 1.

Y-Intercept

Y-intercept (c): The distance from the origin where a line crosses the y-axis.

• Sign convention:
  • Positive if the line crosses above the origin.
  • Negative if the line crosses below the origin.
  • Zero for the x-axis or lines parallel to the y-axis.

Example: For a line crossing the y-axis at (0, 5), the y-intercept c = 5 (positive).

Finding the Slope and the Y-Intercept of a Given Line

• Steps to find slope and y-intercept for a line ax + by + c = 0:
  • Rearrange the equation to the form y = mx + c.
  • Slope (m) = coefficient of x.
  • Y-intercept (c) = constant term.
• Formula:
  • For ax + by + c = 0:
  • Slope (m) = -a/b
  • Y-intercept (c) = -c/b
• Alternate method: Directly identify m and c after converting to y = mx + c.

Example: For 2x - 3y + 5 = 0: 

• Rearrange: -3y = -2x - 5 → y = (2/3)x + 5/3.
• Slope m = 2/3.
• Y-intercept c = 5/3.