The equation of the line whilch bisects the obtuse angle between the lines `x-2y+4=0` and `4x-3y+2=0` is

Options

A. `(4-sqrt(5))x-(3-2sqrt(5))y+(2-4sqrt(5))=0`
B. `(3-2sqrt(5))x-(4-sqrt(5))y+(2+4sqrt(5))=0`
C. `(4+sqrt(5))x-(3+2sqrt(5))y+(2+4sqrt(5))=0`
D. None of these

Answer

To find the equation of the line that bisects the obtuse angle between the lines \( x - 2y + 4 = 0 \) and \( 4x - 3y + 2 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients of the lines
The given lines can be expressed in the standard form \( Ax + By + C = 0 \):
1. For the first line \( x - 2y + 4 = 0 \):
- \( A_1 = 1 \), \( B_1 = -2 \), \( C_1 = 4 \)

2. For the second line \( 4x - 3y + 2 = 0 \):
- \( A_2 = 4 \), \( B_2 = -3 \), \( C_2 = 2 \)

### Step 2: Check the signs of \( A_1, A_2 \) and \( B_1, B_2 \)
We need to check the signs of \( A_1, A_2 \) and \( B_1, B_2 \):
- \( A_1 = 1 \) (positive)
- \( A_2 = 4 \) (positive)
- \( B_1 = -2 \) (negative)
- \( B_2 = -3 \) (negative)

Since both \( A_1 \) and \( A_2 \) are positive, we can proceed to find the angle bisector.

### Step 3: Use the angle bisector formula
The formula for the angle bisector is given by:
\[ \frac{A_1}{\sqrt{A_1^2 + B_1^2}} \cdot (x + \frac{C_1}{A_1}) + \frac{A_2}{\sqrt{A_2^2 + B_2^2}} \cdot (x + \frac{C_2}{A_2}) = 0 \]

Calculating the required values:
- For the first line:
\[ \sqrt{A_1^2 + B_1^2} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

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

### Step 4: Substitute into the angle bisector equation
Substituting the values into the angle bisector equation:
\[ \frac{1}{\sqrt{5}}(x + 4) + \frac{4}{5}(x + \frac{1}{2}) = 0 \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ 5(x + 4) + 4\sqrt{5}(x + \frac{1}{2}) = 0 \]
Expanding this:
\[ 5x + 20 + 4\sqrt{5}x + 2\sqrt{5} = 0 \]
Combining like terms:
\[ (5 + 4\sqrt{5})x + (20 + 2\sqrt{5}) = 0 \]

### Step 6: Rearranging to find the final equation
Rearranging gives us the equation of the obtuse angle bisector:
\[ (5 + 4\sqrt{5})x + (20 + 2\sqrt{5}) = 0 \]

### Final Answer
The equation of the line which bisects the obtuse angle between the lines \( x - 2y + 4 = 0 \) and \( 4x - 3y + 2 = 0 \) is:
\[ (5 + 4\sqrt{5})x + (20 + 2\sqrt{5}) = 0 \]