Chapter Summary

Key Points

• Distance Formula: The distance between two points P(x₁, y₁) and Q(x₂, y₂) is given by:
  • $d = \sqrt{(x₂ - x₁)² + (y₂ - y₁)²}$
• Distance from Origin: The distance of a point P(x, y) from the origin is:
  • $d = \sqrt{x² + y²}$
• Section Formula: The coordinates of the point P(x, y) that divides the line segment joining A(x₁, y₁) and B(x₂, y₂) in the ratio m₁:m₂ are:
  • $x = \frac{m₁x₂ + m₂x₁}{m₁ + m₂}, \quad y = \frac{m₁y₂ + m₂y₁}{m₁ + m₂}$
• Mid-point Formula: The mid-point of the line segment joining A(x₁, y₁) and B(x₂, y₂) is:
  • $P = \left( \frac{x₁ + x₂}{2}, \frac{y₁ + y₂}{2} \right)$

Examples

• Example of Distance Calculation: To find the distance between points (2, 3) and (4, 1), apply the distance formula.
• Example of Section Formula: To find the coordinates of the point dividing the segment joining (-1, 7) and (4, -3) in the ratio 2:3, use the section formula.

Applications

• Coordinate geometry is widely applied in fields such as physics, engineering, navigation, and art.

Chapter 7: Coordinate Geometry

7.1 Introduction

• To locate a point on a plane, a pair of coordinate axes is required.
• The x-coordinate (abscissa) is the distance from the y-axis.
• The y-coordinate (ordinate) is the distance from the x-axis.
• Points on the x-axis are of the form (x, 0) and on the y-axis are of the form (0, y).

7.2 Distance Formula

• The distance between two points P(x₁, y₁) and Q(x₂, y₂) is given by:
  $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• The distance of a point P(x, y) from the origin O(0, 0) is:
  $OP = \sqrt{x^2 + y^2}$

Examples

1. Example 1: Find the distance between points (2, 3) and (4, 1).
2. Example 2: Find the distance between points (0, 0) and (36, 15).

7.3 Section Formula

• The coordinates of the point P(x, y) that divides the line segment joining points A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ are:
  $x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \quad y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}$

Examples

1. Example 6: Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1.
   • Solution: (7, 3)
2. Example 7: Find the ratio in which the point (-4, 6) divides the line segment joining A(-6, 10) and B(3, -8).
   • Solution: m₁ : m₂ = 2 : 7

Exercises

1. Find the distance between the following pairs of points:
   • (2, 3), (4, 1)
   • (-5, 7), (-1, 3)
2. Find the coordinates of the point which divides the join of (-1, 7) and (4, -3) in the ratio 2:3.
3. Determine if the points (1, 5), (2, 3), and (-2, -11) are collinear.

Common Mistakes and Exam Tips

Common Pitfalls

• Misapplication of Distance Formula: Students often forget to square the differences in coordinates when using the distance formula, leading to incorrect calculations.
• Incorrect Use of Section Formula: Failing to correctly apply the section formula can result in wrong coordinates for points dividing line segments.
• Assuming Collinearity: Students may incorrectly assume points are collinear without verifying using the distance formula or slope calculations.
• Ignoring Units: When calculating distances, students sometimes neglect to include units, which can lead to confusion in answers.

Tips for Success

• Double-Check Calculations: Always recheck your calculations for distance and coordinates, especially when applying formulas.
• Visualize Problems: Draw diagrams to help visualize the problem, especially for coordinate geometry questions.
• Practice with Examples: Work through multiple examples of distance and section formula problems to build confidence.
• Understand Concepts: Make sure to understand the underlying concepts of distance and section formulas rather than just memorizing them.