Determining the Equation of a Perpendicular Line

Key Concepts

• Perpendicular Lines: Two lines are perpendicular if they intersect at a right angle (90 degrees). Their slopes are negative reciprocals of each other.
• Negative Reciprocal: To find the negative reciprocal of a slope, flip the fraction and change its sign.

Problem Statement

• Find the equation of a line perpendicular to ( y = 2x + 1 ) and passing through the point ((-4,5)).

Steps to Solve

Step 1: Identify the Slope of the Given Line

• Given line: ( y = 2x + 1 )
• Slope ( (m) = 2/1 )

Step 2: Find the Slope of the Perpendicular Line

• Negative reciprocal of ( 2/1 = -1/2 )

Step 3: Use Slope-Intercept Form

• Slope-Intercept Form: ( y = mx + b )
• Substitute ( m = -1/2 ) into the equation: ( y = -1/2x + b )
• Use the point ((-4,5)) to find ( b ):
  • Substitute ( x = -4 ) and ( y = 5 ) into the equation:
  • [ 5 = -1/2(-4) + b ]
  • Simplifies to ( 5 = 2 + b )
  • Solve for ( b ): ( b = 3 )
• Equation: ( y = -1/2x + 3 )

Step 4: Use Point-Slope Form

• Point-Slope Form: ( y - y_1 = m(x - x_1) )
• Using point ((-4,5)), ( m = -1/2 ):
  • ( y - 5 = -1/2(x + 4) )
  • Distribute and simplify:
    • ( y - 5 = -1/2x - 2 )
  • Add 5 to both sides:
    • ( y = -1/2x + 3 )
• Result: Same equation ( y = -1/2x + 3 )

Verification

• Graphically:
  • Original line: ( y = 2x + 1 )
  • Perpendicular line: ( y = -1/2x + 3 )
  • Lines intersect at a right angle
  • Perpendicular line passes through ((-4,5))

Conclusion

• The equation of the line perpendicular to ( y = 2x + 1 ) and passing through ((-4,5)) is ( y = -1/2x + 3 ).
• Method verified using both slope-intercept and point-slope forms.