Lecture on Perpendicular Lines

Key Concepts

• Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -1.
• Slope Formula: If line equations are given as:
  • (y = m_1x + b_1)
  • (y = m_2x + b_2)
  • Then (m_1 \times m_2 = -1)

Finding Slopes

• Example Given Line: (y = 2x + 5)
  • Given slope (m = 2) (expressed as (\frac{2}{1}))
  • Perpendicular slope is the negative reciprocal: (-\frac{1}{2})

Finding Equation of Perpendicular Line

• Using Point-Slope Formula:
  • Example point: ((0, 0))
  • Equation: (y - 0 = -\frac{1}{2}(x - 4))
  • Simplified: (y = -\frac{1}{2}x + 2)

Equation in Standard Form

• Example: (3x + 5y = 10)
  • Solve for y to find slope:
    • Rearrange: (5y = -3x + 10)
    • (y = -\frac{3}{5}x + 2)
  • Perpendicular slope: (\frac{5}{3})

Using Point to Find Perpendicular Line

• Example Point: ((3, -7))
  • Equation: (y + 7 = \frac{5}{3}(x - 3))
  • Distribute and simplify: (y = \frac{5}{3}x - 12)
  • Convert to standard form:
    • Multiply by 3: (3y = 5x - 36)
    • Rearrange: (-5x + 3y = -36)
    • Simplify by multiplying through by (-1): (5x - 3y = 36)

Standard Form Switching

• Concept: Given (ax + by = c), a perpendicular line can be (bx - ay = d).
• Example:
  • Given: (4x - 3y = -6)
  • Find perpendicular line through ((2, 1))
  • Switch coefficients and signs: (3x + 4y = d)
  • Plug in point ((2, 1)):
    • (3 \times 2 + 4 \times 1 = 10)
  • Final Equation: (3x + 4y = 10)

Summary

• Perpendicular lines have slopes that are negative reciprocals.
• Point-slope and standard form methods are used to find equations of perpendicular lines.
• Switching coefficients in standard form helps in quickly finding perpendicular lines.