Find the equations to the straight lines bisecting the angles between the following pairs of straight lines, placing first the bisector of the angle in which the origin lies.
`12x + 5y-4= 0 and 3x+ 4y+ 7=0`

Answer

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

### Step 1: Identify the given lines
The equations of the lines are:
1. \( L_1: 12x + 5y - 4 = 0 \)
2. \( L_2: 3x + 4y + 7 = 0 \)

### Step 2: Calculate \( L_1(0,0) \) and \( L_2(0,0) \)
Evaluate both equations at the origin (0,0):
- For \( L_1 \):
\[ L_1(0,0) = 12(0) + 5(0) - 4 = -4 \]
- For \( L_2 \):
\[ L_2(0,0) = 3(0) + 4(0) + 7 = 7 \]

### Step 3: Check the product of \( L_1(0,0) \) and \( L_2(0,0) \)
Calculate the product:
\[ L_1(0,0) \cdot L_2(0,0) = (-4) \cdot 7 = -28 \]
Since the product is less than 0, the origin lies between the two lines.

### Step 4: Use the angle bisector formula
The angle bisector formula for two lines \( L_1 \) and \( L_2 \) is given by:
\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]
Where:
- \( A_1 = 12, B_1 = 5, C_1 = -4 \)
- \( A_2 = 3, B_2 = 4, C_2 = 7 \)

### Step 5: Calculate the denominators
Calculate the square roots:
\[ \sqrt{A_1^2 + B_1^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
\[ \sqrt{A_2^2 + B_2^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 6: Write the equations for the angle bisectors
Using the angle bisector formula:
1. For the bisector where the origin lies:
\[ \frac{12x + 5y - 4}{13} = -\frac{3x + 4y + 7}{5} \]
Cross-multiplying gives:
\[ 5(12x + 5y - 4) = -13(3x + 4y + 7) \]
Expanding both sides:
\[ 60x + 25y - 20 = -39x - 52y - 91 \]
Combining like terms:
\[ 99x + 77y + 71 = 0 \]

2. For the bisector where the origin does not lie:
\[ \frac{12x + 5y - 4}{13} = \frac{3x + 4y + 7}{5} \]
Cross-multiplying gives:
\[ 5(12x + 5y - 4) = 13(3x + 4y + 7) \]
Expanding both sides:
\[ 60x + 25y - 20 = 39x + 52y + 91 \]
Combining like terms:
\[ 7x - 9y - 37 = 0 \]

### Final Equations
The equations of the angle bisectors are:
1. \( 99x + 77y + 71 = 0 \) (bisector where the origin lies)
2. \( 7x - 9y - 37 = 0 \) (bisector where the origin does not lie)