Find the equations of the bisectors of the angles formed by the lines :
`3x-4y+12=0` and `4x+3y+2=0`

Answer

To find the equations of the bisectors of the angles formed by the lines \(3x - 4y + 12 = 0\) and \(4x + 3y + 2 = 0\), we will use the formula for the angle bisectors of two lines given in the form \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\).

### Step 1: Identify coefficients
From the equations of the lines, we can identify:
- For the first line \(3x - 4y + 12 = 0\):
- \(a_1 = 3\)
- \(b_1 = -4\)
- \(c_1 = 12\)

- For the second line \(4x + 3y + 2 = 0\):
- \(a_2 = 4\)
- \(b_2 = 3\)
- \(c_2 = 2\)

### Step 2: Use the angle bisector formula
The equations of the angle bisectors 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 denominators
Calculate \(\sqrt{a_1^2 + b_1^2}\) and \(\sqrt{a_2^2 + b_2^2}\):
- \(\sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)
- \(\sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\)

### Step 4: Substitute into the bisector formula
Substituting the values into the bisector formula gives:
\[ \frac{3x - 4y + 12}{5} = \pm \frac{4x + 3y + 2}{5} \]

### Step 5: Simplify the equations
1. For the positive case:
\[ 3x - 4y + 12 = 4x + 3y + 2 \]
Rearranging gives:
\[ 3x - 4x - 4y - 3y + 12 - 2 = 0 \]
Simplifying:
\[ -x + 7y + 10 = 0 \quad \Rightarrow \quad x - 7y - 10 = 0 \]

2. For the negative case:
\[ 3x - 4y + 12 = - (4x + 3y + 2) \]
Rearranging gives:
\[ 3x - 4y + 12 + 4x + 3y + 2 = 0 \]
Simplifying:
\[ 7x - y + 14 = 0 \]

### Final Result
The equations of the angle bisectors are:
1. \(x - 7y - 10 = 0\)
2. \(7x - y + 14 = 0\)