The bisector of the acute angle formed between the lines 4x - 3y + 7 = 0 and 3x - 4y + 14 = 0 has the equation

Options

A. x + y + 3 = 0
B. x - y - 3 = 0
C. x - y + 3 =0
D. 3x - y - 7 = 0

Explanation

To find the equation of the bisector of the acute angle formed between the lines \(4x - 3y + 7 = 0\) and \(3x - 4y + 14 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
From the equations of the lines, we can identify the coefficients:
- For the first line \(4x - 3y + 7 = 0\):
- \(A_1 = 4\), \(B_1 = -3\), \(C_1 = 7\)

- For the second line \(3x - 4y + 14 = 0\):
- \(A_2 = 3\), \(B_2 = -4\), \(C_2 = 14\)

### Step 2: Write the equations for the angle bisectors
The equations of the angle bisectors can be expressed as:
\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]

### Step 3: Calculate the magnitudes of the coefficients
Calculate the magnitudes:
- For the first line:
\[ \sqrt{A_1^2 + B_1^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

- For the second line:
\[ \sqrt{A_2^2 + B_2^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Substitute into the angle bisector equation
Substituting into the angle bisector equation gives:
\[ \frac{4x - 3y + 7}{5} = \pm \frac{3x - 4y + 14}{5} \]

### Step 5: Simplify the equation
This can be simplified by multiplying through by 5:
\[ 4x - 3y + 7 = \pm (3x - 4y + 14) \]

This leads to two cases:
1. \(4x - 3y + 7 = 3x - 4y + 14\)
2. \(4x - 3y + 7 = -(3x - 4y + 14)\)

### Step 6: Solve the first case
For the first case:
\[ 4x - 3y + 7 = 3x - 4y + 14 \]
Rearranging gives:
\[ 4x - 3x + 4y - 3y + 7 - 14 = 0 \]
\[ x + y - 7 = 0 \quad \text{or} \quad x + y = 7 \]

### Step 7: Solve the second case
For the second case:
\[ 4x - 3y + 7 = -3x + 4y - 14 \]
Rearranging gives:
\[ 4x + 3x - 3y - 4y + 7 + 14 = 0 \]
\[ 7x - 7y + 21 = 0 \quad \text{or} \quad x - y + 3 = 0 \quad \text{or} \quad x - y = -3 \]

### Step 8: Conclusion
The equations of the angle bisectors are:
1. \(x + y = 7\)
2. \(x - y = -3\)

Since we need the acute angle bisector, we can choose one of the equations based on the context of the problem.

The final answer is:
- The equation of the bisector of the acute angle formed between the lines is \(x + y = 7\).