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
For the lines given:
1. Line 1: \(4x - 3y + 7 = 0\) (Here, \(A_1 = 4\), \(B_1 = -3\), \(C_1 = 7\))
2. Line 2: \(3x - 4y + 14 = 0\) (Here, \(A_2 = 3\), \(B_2 = -4\), \(C_2 = 14\))

### Step 2: Check the sign of \(A_1A_2 + B_1B_2\)
Calculate:
\[ A_1A_2 + B_1B_2 = (4)(3) + (-3)(-4) = 12 + 12 = 24 \]
Since \(24 > 0\), the angle bisector we are looking for corresponds to the acute angle.

### Step 3: Write the equation of the angle bisector
The equation of the angle bisector can be expressed as:
\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]
Substituting the values:
\[ \frac{4x - 3y + 7}{\sqrt{4^2 + (-3)^2}} = \frac{3x - 4y + 14}{\sqrt{3^2 + (-4)^2}} \]
Calculating the denominators:
\[ \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
\[ \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, the equation simplifies to:
\[ \frac{4x - 3y + 7}{5} = \pm \frac{3x - 4y + 14}{5} \]
Multiplying through by 5 gives:
\[ 4x - 3y + 7 = \pm (3x - 4y + 14) \]

### Step 4: Solve for the two cases
**Case 1:**
\[ 4x - 3y + 7 = 3x - 4y + 14 \]
Rearranging gives:
\[ 4x - 3x + 4y - 3y + 7 - 14 = 0 \implies x + y - 7 = 0 \implies x + y = 7 \]

**Case 2:**
\[ 4x - 3y + 7 = - (3x - 4y + 14) \]
Rearranging gives:
\[ 4x - 3y + 7 + 3x - 4y - 14 = 0 \implies 7x - y - 7 = 0 \implies 7x - y = 7 \implies y = 7x - 7 \]

### Step 5: Identify the acute angle bisector
The acute angle bisector is the one that corresponds to the positive case, which is:
\[ x + y - 7 = 0 \]

### Final Answer
The equation of the bisector of the acute angle formed between the lines is:
\[ x + y - 7 = 0 \]