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

Options

A. `21x+77y-101=0`
B. `11x+3y+20=0`
C. `21x-7y+3=0`
D. `11x-3y+9=0`

Answer

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

### Step 1: Rewrite the equations in standard form
The equations are already in the standard form \(Ax + By + C = 0\):
1. Line 1: \(3x - 4y + 7 = 0\) (Here, \(A_1 = 3\), \(B_1 = -4\), \(C_1 = 7\))
2. Line 2: \(12x + 5y - 2 = 0\) (Here, \(A_2 = 12\), \(B_2 = 5\), \(C_2 = -2\))

### Step 2: Make the constant term of one equation positive
To ensure consistency, we can multiply the second equation by -1 to make the constant term positive:
\[ - (12x + 5y - 2) = 0 \implies -12x - 5y + 2 = 0 \]

### Step 3: Check the condition for acute angle bisector
We need to check if \(A_1A_2 + B_1B_2 < 0\):
\[ A_1A_2 + B_1B_2 = 3 \cdot 12 + (-4) \cdot 5 = 36 - 20 = 16 \]
Since \(16 > 0\), we will use the positive sign for the acute angle bisector formula.

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

Calculating the denominators:
\[ \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
\[ \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 5: Set up the equation
Thus, we have:
\[ \frac{3x - 4y + 7}{5} = \frac{12x + 5y - 2}{13} \]

### Step 6: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 13(3x - 4y + 7) = 5(12x + 5y - 2) \]

### Step 7: Expand both sides
Expanding both sides:
\[ 39x - 52y + 91 = 60x + 25y - 10 \]

### Step 8: Rearrange the equation
Bringing all terms to one side:
\[ 39x - 60x - 52y - 25y + 91 + 10 = 0 \]
\[ -21x - 77y + 101 = 0 \]
Multiplying through by -1 gives:
\[ 21x + 77y - 101 = 0 \]

### Step 9: Simplify the equation
To simplify, we can divide by the common factor (if any). However, in this case, we can leave it as is or check for further simplification.

### Final Equation
The equation of the acute angle bisector is:
\[ 21x + 77y - 101 = 0 \]