Find the equations of the bisectors of the angle between the straight line 3x + 4y + 2 = 0 and 5x - 12y - 6 = 0

Options

A. 8x + y + 7 = 0
B. 16x - 12y -1 = 0
C. x + 8y + 4 = 0
D. BOTH (A) AND (B)

Answer

To find the equations of the bisectors of the angle between the straight lines given by the equations \(3x + 4y + 2 = 0\) and \(5x - 12y - 6 = 0\), we can use the angle bisector formula.

### Step-by-step Solution:

1. **Identify the coefficients:**
For the first line \(3x + 4y + 2 = 0\):
- \(A_1 = 3\)
- \(B_1 = 4\)
- \(C_1 = 2\)

For the second line \(5x - 12y - 6 = 0\):
- \(A_2 = 5\)
- \(B_2 = -12\)
- \(C_2 = -6\)

2. **Use the angle bisector formula:**
The formula for the angle bisector between two lines is given by:
\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]

3. **Calculate the denominators:**
- For the first line:
\[ \sqrt{A_1^2 + B_1^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- For the second line:
\[ \sqrt{A_2^2 + B_2^2} = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

4. **Set up the equation:**
Substitute the values into the angle bisector formula:
\[ \frac{3x + 4y + 2}{5} = \pm \frac{5x - 12y - 6}{13} \]

5. **Cross-multiply to eliminate the fractions:**
- For the positive case:
\[ 13(3x + 4y + 2) = 5(5x - 12y - 6) \]
- For the negative case:
\[ 13(3x + 4y + 2) = -5(5x - 12y - 6) \]

6. **Solve the positive case:**
Expanding the equation:
\[ 39x + 52y + 26 = 25x - 60y - 30 \]
Rearranging gives:
\[ 39x - 25x + 52y + 60y + 26 + 30 = 0 \]
Simplifying:
\[ 14x + 112y + 56 = 0 \]
Dividing by 14:
\[ x + 8y + 4 = 0 \]

7. **Solve the negative case:**
Expanding the equation:
\[ 39x + 52y + 26 = -25x + 60y + 30 \]
Rearranging gives:
\[ 39x + 25x + 52y - 60y + 26 - 30 = 0 \]
Simplifying:
\[ 64x - 8y - 4 = 0 \]
Dividing by 4:
\[ 16x - 2y - 1 = 0 \]

8. **Final equations of the angle bisectors:**
The equations of the angle bisectors are:
1. \(x + 8y + 4 = 0\)
2. \(16x - 2y - 1 = 0\)