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-7=0`
B. `x-y+3=0`
C. `2x+y-11=0`
D. `x+2y-12=0`

Answer

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
The equations of the lines can be expressed in the standard form \(A_1x + B_1y + C_1 = 0\) and \(A_2x + B_2y + C_2 = 0\).

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: Use the angle bisector formula
The formula for the angle bisector between two lines is given by:

\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]

### Step 3: Calculate the square roots
First, we calculate the denominators:

\[ \sqrt{A_1^2 + B_1^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

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

### Step 4: Substitute into the formula
Now substituting into the angle bisector formula:

\[ \frac{4x - 3y + 7}{5} = \pm \frac{3x - 4y + 14}{5} \]

### Step 5: Simplify the equation
Since both sides are divided by 5, we can multiply through by 5 to eliminate the denominator:

\[ 4x - 3y + 7 = \pm (3x - 4y + 14) \]

This gives us two cases to consider:

**Case 1:**
\[ 4x - 3y + 7 = 3x - 4y + 14 \]

Rearranging gives:

\[ 4x - 3x + 4y - 3y + 7 - 14 = 0 \]
\[ x + y - 7 = 0 \quad \text{(1)} \]

**Case 2:**
\[ 4x - 3y + 7 = -(3x - 4y + 14) \]

Rearranging gives:

\[ 4x - 3y + 7 = -3x + 4y - 14 \]
\[ 4x + 3x - 3y - 4y + 7 + 14 = 0 \]
\[ 7x - 7y + 21 = 0 \]
\[ x - y + 3 = 0 \quad \text{(2)} \]

### Step 6: Identify the acute angle bisector
To determine which of these represents the acute angle bisector, we check the sign of \(A_1A_2 + B_1B_2\):

\[ A_1A_2 + B_1B_2 = 4 \cdot 3 + (-3)(-4) = 12 + 12 = 24 > 0 \]

Since this is positive, we use the negative sign in the angle bisector formula, which corresponds to equation (2):

\[ x - y + 3 = 0 \]

### Final Answer
The equation of the bisector of the acute angle formed between the lines is:

\[ x - y + 3 = 0 \]