Combined Equation of Angle Bisectors of Pair of Lines

Combined Equation of Angle Bisectors of Pair of Lines:

1. Angle Bisectors:

  • For a pair of lines represented by ax2+2hxy+by2+2gx+2fy+c=0 and px2+2qxy+ry2+2sx+2ty+u=0, the equation of the angle bisectors can be found.

2. Combined Equation of Angle Bisectors:

  • The combined equation of the angle bisectors of a pair of lines is given by: S1x2+2T1xy+S2y2+2T2x+2T2y+S3=0 where: S1=a+b2ab

S2=p+r2pr

S3=2(gu+fv)(gs+ft)2

 T1=abgf

 T2=prst

Example:

Given two lines:

  • 2x2+3xy2y2+4x6y5=0
  • 3x24xy+y2+2x+3y1=0

Find the combined equation of their angle bisectors.

Solution:

Comparing the given equations with the standard form ax2+2hxy+by2+2gx+2fy+c=0:

  • For the first equation: a=2, h=32, b=2, g=2, f=3, c=5
  • For the second equation: a=3, h=2, b=1, g=2, f=3, c=1

Using the formulas for S1, S2, S3, T1, and T2: S1=a+b2ab=2222(2)=424=44=8S2=a+b2ab=3+1231=423 S3=2(gu+fv)(gs+ft)2=2(2(1)+(3)(3))(22+(3)3)2 S3=2(2+9)(49)2=2(7)(5)2=1425=11T2=abgf=(3)(1)23=35

Hence, the combined equation of the angle bisectors will be: 8x2+(2i+1)xy+(423)y2+23x5y11=0

This equation represents the combined equation of the angle bisectors of the given pair of lines.

3. Derivation of the Combined Equation:

  • Obtained by finding the equation of angle bisectors using the concept of internal and external angle bisectors of a pair of lines.

4. Explanation and Interpretation:

  • S1x2+2T1xy+S2y2+2T2x+2T2y+S3=0 represents the equation of the angle bisectors of the given pair of lines.
  • S1 and S2 are the coefficients of the terms x2 and y2 respectively, representing the combined effect of the original coefficients.
  • S3 involves the mixed terms and constants from the original equations.
  • T1 and T2 represent the coefficients of xy terms in the combined equation.

5. Applications and Usage:

  • Geometry: Understanding the orientation and characteristics of the angle bisectors.
  • Problem Solving: Utilizing the combined equation in various problems related to the angle bisectors of lines.
  • Mathematical Analysis: Helps in studying the relationship between lines and their angle bisectors.