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

Options

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

Answer

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

### Step 1: Identify the coefficients of the lines
The equations of the lines are:
1. \(L_1: x - 2y + 4 = 0\)
2. \(L_2: 4x - 3y + 2 = 0\)

From these equations, we can identify:
- For \(L_1\): \(A_1 = 1\), \(B_1 = -2\), \(C_1 = 4\)
- For \(L_2\): \(A_2 = 4\), \(B_2 = -3\), \(C_2 = 2\)

### Step 2: Calculate the angle bisector formula
The angle bisectors of two lines can be found using the formula:
\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]

### Step 3: Calculate the left-hand side for both lines
For \(L_1\):
\[ \sqrt{A_1^2 + B_1^2} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]
Thus, the left-hand side becomes:
\[ \frac{1 \cdot x - 2 \cdot y + 4}{\sqrt{5}} = \frac{x - 2y + 4}{\sqrt{5}} \]

For \(L_2\):
\[ \sqrt{A_2^2 + B_2^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
Thus, the right-hand side becomes:
\[ \frac{4 \cdot x - 3 \cdot y + 2}{5} \]

### Step 4: Set up the equation for the obtuse angle bisector
Since we want the obtuse angle bisector, we take the positive sign in the equation:
\[ \frac{x - 2y + 4}{\sqrt{5}} = \frac{4x - 3y + 2}{5} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 5(x - 2y + 4) = \sqrt{5}(4x - 3y + 2) \]

### Step 6: Expand both sides
Expanding the left-hand side:
\[ 5x - 10y + 20 \]
Expanding the right-hand side:
\[ 4\sqrt{5}x - 3\sqrt{5}y + 2\sqrt{5} \]

### Step 7: Rearranging the equation
Setting both sides equal gives:
\[ 5x - 10y + 20 = 4\sqrt{5}x - 3\sqrt{5}y + 2\sqrt{5} \]
Rearranging terms leads to:
\[ 5x - 4\sqrt{5}x + 3\sqrt{5}y - 10y + 20 - 2\sqrt{5} = 0 \]

### Step 8: Combine like terms
Combining like terms yields:
\[ (5 - 4\sqrt{5})x + (3\sqrt{5} - 10)y + (20 - 2\sqrt{5}) = 0 \]

### Final Step: Simplify the equation
This represents the equation of the line that bisects the obtuse angle between the two given lines.