# 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 $a{x}^{2}+2hxy+b{y}^{2}+2gx+2fy+c=0$ and $p{x}^{2}+2qxy+r{y}^{2}+2sx+2ty+u=0$, 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: ${S}_{1}{x}^{2}+2{T}_{1}xy+{S}_{2}{y}^{2}+2{T}_{2}x+2{T}_{2}y+{S}_{3}=0$ where: ${S}_{1}=a+b-2\sqrt{ab}$

### Example:

Given two lines:

• $2{x}^{2}+3xy-2{y}^{2}+4x-6y-5=0$
• $3{x}^{2}-4xy+{y}^{2}+2x+3y-1=0$

Find the combined equation of their angle bisectors.

Solution:

Comparing the given equations with the standard form $a{x}^{2}+2hxy+b{y}^{2}+2gx+2fy+c=0$:

• For the first equation: $a=2$, $h=\frac{3}{2}$, $b=-2$, $g=2$, $f=-3$, $c=-5$
• For the second equation: $a=3$, $h=-2$, $b=1$, $g=2$, $f=3$, $c=-1$

Using the formulas for ${S}_{1}$, ${S}_{2}$, ${S}_{3}$, ${T}_{1}$, and ${T}_{2}$: ${S}_{1}=a+b-2\sqrt{ab}=2-2-2\sqrt{2\cdot \left(-2\right)}=-4-2\sqrt{4}=-4-4=-8$${S}_{2}=a+b-2\sqrt{ab}=3+1-2\sqrt{3\cdot 1}=4-2\sqrt{3}$ ${S}_{3}=2\left(gu+fv\right)-\left(gs+ft{\right)}^{2}=2\left(2\cdot \left(-1\right)+\left(-3\right)\cdot \left(-3\right)\right)-\left(2\cdot 2+\left(-3\right)\cdot 3{\right)}^{2}$ ${S}_{3}=2\left(-2+9\right)-\left(4-9{\right)}^{2}=2\left(7\right)-\left(-5{\right)}^{2}=14-25=-11$${T}_{2}=\sqrt{ab}-g-f=\sqrt{\left(3\right)\left(1\right)}-2-3=\sqrt{3}-5$

Hence, the combined equation of the angle bisectors will be: $-8{x}^{2}+\left(2i+1\right)xy+\left(4-2\sqrt{3}\right){y}^{2}+2\sqrt{3}x-5y-11=0$

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:

• ${S}_{1}{x}^{2}+2{T}_{1}xy+{S}_{2}{y}^{2}+2{T}_{2}x+2{T}_{2}y+{S}_{3}=0$ represents the equation of the angle bisectors of the given pair of lines.
• ${S}_{1}$ and ${S}_{2}$ are the coefficients of the terms ${x}^{2}$ and ${y}^{2}$ respectively, representing the combined effect of the original coefficients.
• ${S}_{3}$ involves the mixed terms and constants from the original equations.
• ${T}_{1}$ and ${T}_{2}$ 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.