# 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: ${S}_{1}=a+b-2\sqrt{ab}$

### ${S}_{2}=p+r-2\sqrt{pr}$

### ${S}_{3}=2(gu+fv)-(gs+ft{)}^{2}$

### ${T}_{1}=\sqrt{ab}-g-f$

### ${T}_{2}=\sqrt{pr}-s-t$

### 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 (-2)}=-4-2\sqrt{4}=-4-4=-8$${S}_{2}=a+b-2\sqrt{ab}=3+1-2\sqrt{3\cdot 1}=4-2\sqrt{3}$ ${S}_{3}=2(gu+fv)-(gs+ft{)}^{2}=2(2\cdot (-1)+(-3)\cdot (-3))-(2\cdot 2+(-3)\cdot 3{)}^{2}$ ${S}_{3}=2(-2+9)-(4-9{)}^{2}=2(7)-(-5{)}^{2}=14-25=-11$${T}_{2}=\sqrt{ab}-g-f=\sqrt{(3)(1)}-2-3=\sqrt{3}-5$

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

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.