Lecture Notes on Coordinate Geometry and Straight Lines

Introduction

• Welcome and session overview
• Today's session focuses on basics of coordinate geometry and straight lines.

Agenda

1. Basics of Coordinate Geometry
   • Distance Formula
   • Section Formula
   • Important Centers related to Triangle
2. Locus-related problems
3. Shifting of Origin

Instructor Introduction

• Instructor with over 10 years of experience in teaching.
• Successful track record of students in IITs, NITs, etc.

Straight Lines Overview

• Importance of understanding straight lines for further chapters:
  • Circles
  • Conic sections (Parabola, Ellipse, Hyperbola)
  • 3D Geometry

Coordinate Plane Basics

• X-axis and Y-axis orientation and quadrants:
  • First Quadrant: (x > 0, y > 0)
  • Second Quadrant: (x < 0, y > 0)
  • Third Quadrant: (x < 0, y < 0)
  • Fourth Quadrant: (x > 0, y < 0)

Distance Formula

• Distance between two points P(x1, y1) and Q(x2, y2):
  • Formula:
    [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
• Example Problem: Calculate distance between (7, -6) and (-3, 4).
• Result: 10√2 units.

Section Formula

• Internal Division:
  • Given two points A(x1, y1) and B(x2, y2), a point P divides the line segment in the ratio m:n.
  • Coordinates of P:
    [ P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) ]
• Midpoint Formula:
  • Midpoint of segment AB:
    [ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]

Centers of a Triangle

Centroid

• Coordinates of centroid (G):
  • [ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) ]
• Centroid divides medians in the ratio 2:1.

Incenter

• Incenter (I) is the intersection of angle bisectors.
• Coordinates:
  • [ I = \left( \frac{a x_1 + b x_2 + c x_3}{a+b+c}, \frac{a y_1 + b y_2 + c y_3}{a+b+c} \right) ]

Orthocenter

• Orthocenter (H) is the intersection of altitudes in a triangle.
• Special case for right triangle: the orthocenter is at the vertex with a right angle.

Circumcenter

• Circumcenter (C) is the intersection of perpendicular bisectors.
• In a right triangle, circumcenter is at the midpoint of the hypotenuse.

Locus

Definition

• Locus is the path traced by a moving point that satisfies a given condition.

Steps to Find Locus

1. Define the Moving Point: Assign coordinates (h, k)
2. Write the Condition: Convert the geometric condition into a mathematical form.
3. Eliminate Variables: Replace h and k with x and y to get the final equation.

Example Problems

1. Find the locus of a point equidistant from two points A and B.
2. Midpoint of a rod sliding on two perpendicular lines.

Shifting of Origin

• The process of translating the coordinate system to a new origin (h, k).
• Original coordinates are related to the new coordinates by:
  • [ x = X + h, y = Y + k ]

Examples

1. Shifting origin to eliminate constant terms in equations.
2. Finding transform equations based on new origin.

Conclusion

• Next session will cover advanced topics related to straight lines.
• Emphasis on importance of foundational concepts in geometry for solving complex problems.