Chapter: Straight Lines

• Introduction & Prerequisites

  • Coordinate Geometry Basics
  • Distance Formula between two points
  • Section Formula (Internal & External division)
  • Mid-point Formula
  • Area of a Triangle using vertices

• Slope (Gradient) of a Line ($m$)

  • Definition: $\tan\theta$ where $\theta$ is the inclination
  • Formula given two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • Conditions for Lines
    • Parallel lines: $m_1 = m_2$
    • Perpendicular lines: $m_1 \cdot m_2 = -1$
  • Angle between two lines: $\tan\theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$
  • Collinearity of three points (Slopes of segments are equal)

• Various Forms of the Equation of a Line

  • Slope-Intercept Form: $y = mx + c$
  • Point-Slope Form: $y - y_1 = m(x - x_1)$
  • Two-Point Form: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
  • Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$
  • Normal Form: $x\cos\alpha + y\sin\alpha = p$

• General Equation of a Line

  • Standard Form: $Ax + By + C = 0$
  • Conversion to Slope-Intercept Form: $y = -\left(\frac{A}{B}\right)x - \frac{C}{B}$
    • Slope $m = -\frac{A}{B}$
    • Y-intercept $= -\frac{C}{B}$
  • Conversion to Intercept Form: $\frac{x}{-C/A} + \frac{y}{-C/B} = 1$
  • Conversion to Normal Form (using $\sqrt{A^2 + B^2}$)

• Distance of a Point from a Line

  • Perpendicular distance formula from point $(x_1, y_1)$ to line $Ax + By + C = 0$:
    • $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$
  • Distance between Parallel Lines ($Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$):
    • $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$