Find the equations of the lines bisecting the angles between the following pairs of straight lines writing first the bisector of the angle in which the origin lies :
`3x-4y+10=0, 5x-12y-10=0`

Answer

To find the equations of the lines bisecting the angles between the given pairs of straight lines, we will follow these steps:

### Given Lines:
1. \( L_1: 3x - 4y + 10 = 0 \)
2. \( L_2: 5x - 12y - 10 = 0 \)

### Step 1: Identify coefficients
For the equations of the lines, we can identify:
- For \( L_1 \): \( a_1 = 3, b_1 = -4, c_1 = 10 \)
- For \( L_2 \): \( a_2 = 5, b_2 = -12, c_2 = -10 \)

### Step 2: Check the position of the origin
We will evaluate both lines at the origin (0, 0):
- For \( L_1 \):
\[ L_1(0, 0) = 3(0) - 4(0) + 10 = 10 \]
- For \( L_2 \):
\[ L_2(0, 0) = 5(0) - 12(0) - 10 = -10 \]

### Step 3: Determine the product of evaluations
Now, we calculate the product:
\[ L_1(0, 0) \cdot L_2(0, 0) = 10 \cdot (-10) = -100 \]
Since the product is negative, it indicates that the origin lies between the two lines.

### Step 4: Write the equation for the angle bisectors
The equations for the angle bisectors can be given as:
\[ \frac{a_1 x + b_1 y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2 x + b_2 y + c_2}{\sqrt{a_2^2 + b_2^2}} \]

### Step 5: Calculate the denominators
Calculate the denominators:
- For \( L_1 \):
\[ \sqrt{a_1^2 + b_1^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- For \( L_2 \):
\[ \sqrt{a_2^2 + b_2^2} = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

### Step 6: Write the equations for the bisectors
Using the values calculated:
1. For the bisector where the origin lies (using the negative sign):
\[ \frac{3x - 4y + 10}{5} = -\frac{5x - 12y - 10}{13} \]

Cross-multiplying gives:
\[ 13(3x - 4y + 10) = -5(5x - 12y - 10) \]
Expanding both sides:
\[ 39x - 52y + 130 = -25x + 60y + 50 \]

Rearranging gives:
\[ 39x + 25x - 52y - 60y + 130 - 50 = 0 \]
\[ 64x - 112y + 80 = 0 \]

### Step 7: Simplify the equation
Dividing through by 16:
\[ 4x - 7y + 5 = 0 \quad \text{(Equation 1)} \]

2. For the other bisector (using the positive sign):
\[ \frac{3x - 4y + 10}{5} = \frac{5x - 12y - 10}{13} \]

Cross-multiplying gives:
\[ 13(3x - 4y + 10) = 5(5x - 12y - 10) \]
Expanding both sides:
\[ 39x - 52y + 130 = 25x - 60y - 50 \]

Rearranging gives:
\[ 39x - 25x + 60y - 52y + 130 + 50 = 0 \]
\[ 14x + 8y + 180 = 0 \]

### Step 8: Simplify the equation
Dividing through by 2:
\[ 7x + 4y + 90 = 0 \quad \text{(Equation 2)} \]

### Final Result
The equations of the angle bisectors are:
1. \( 4x - 7y + 5 = 0 \)
2. \( 7x + 4y + 90 = 0 \)