Find the equation of the bisector of the acute angle between the lines `3x-4y+7=0 and 12x+5y-2=0`.

Answer

To find the equation of the bisector of the acute angle between the lines \(3x - 4y + 7 = 0\) and \(12x + 5y - 2 = 0\), we can follow these steps:

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

### Step 2: Calculate the coefficients
From the equations, we identify the coefficients:
- For \(L_1\): \(A_1 = 3\), \(B_1 = -4\), \(C_1 = 7\)
- For \(L_2\): \(A_2 = 12\), \(B_2 = 5\), \(C_2 = -2\)

### Step 3: Use the formula for the angle bisectors
The equations of the angle bisectors of the lines can be given by the formula:
\[ \frac{A_1 x + B_1 y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2 x + B_2 y + C_2}{\sqrt{A_2^2 + B_2^2}} \]

### Step 4: Calculate the denominators
Calculate the square roots of the sums of the squares of the coefficients:
- For \(L_1\):
\[ \sqrt{A_1^2 + B_1^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- For \(L_2\):
\[ \sqrt{A_2^2 + B_2^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 5: Set up the equations
Substituting the values into the angle bisector formula gives us:
\[ \frac{3x - 4y + 7}{5} = \pm \frac{12x + 5y - 2}{13} \]

### Step 6: Solve for the positive case
First, consider the positive case:
\[ \frac{3x - 4y + 7}{5} = \frac{12x + 5y - 2}{13} \]
Cross-multiplying gives:
\[ 13(3x - 4y + 7) = 5(12x + 5y - 2) \]
Expanding both sides:
\[ 39x - 52y + 91 = 60x + 25y - 10 \]
Rearranging terms:
\[ 39x - 60x - 52y - 25y + 91 + 10 = 0 \]
This simplifies to:
\[ -21x - 77y + 101 = 0 \quad \text{or} \quad 21x + 77y - 101 = 0 \]

### Step 7: Solve for the negative case
Now consider the negative case:
\[ \frac{3x - 4y + 7}{5} = -\frac{12x + 5y - 2}{13} \]
Cross-multiplying gives:
\[ 13(3x - 4y + 7) = -5(12x + 5y - 2) \]
Expanding both sides:
\[ 39x - 52y + 91 = -60x - 25y + 10 \]
Rearranging terms:
\[ 39x + 60x - 52y + 25y + 91 - 10 = 0 \]
This simplifies to:
\[ 99x - 27y + 81 = 0 \quad \text{or} \quad 11x - 3y + 9 = 0 \]

### Step 8: Determine the acute angle bisector
To determine which bisector corresponds to the acute angle, we can find the slopes of the lines and the bisectors.

- Slope of \(L_1\): \(m_1 = \frac{3}{4}\)
- Slope of \(L_2\): \(m_2 = -\frac{12}{5}\)
- Slope of the first bisector \(21x + 77y - 101 = 0\) is \(-\frac{21}{77}\)
- Slope of the second bisector \(11x - 3y + 9 = 0\) is \(\frac{11}{3}\)

### Step 9: Calculate the angle between the lines and the bisectors
Using the formula for the angle between two lines, we can determine which bisector corresponds to the acute angle. After calculating, we find that the bisector \(11x - 3y + 9 = 0\) corresponds to the acute angle.

### Final Answer
Thus, the equation of the bisector of the acute angle between the lines is:
\[ \boxed{11x - 3y + 9 = 0} \]