A triangle is formed by the lines whose equations are `3x+4y-6=0,12x-5y-3 = 0 and 4x-3y + 12 = 0`. Find the internal bisector of the angle opposite to the side `3x + 4y -6=0.`

Answer

To find the internal bisector of the angle opposite to the side defined by the line \(3x + 4y - 6 = 0\), we will follow these steps:

### Step 1: Identify the lines forming the triangle
The triangle is formed by the following lines:
1. \(L_1: 3x + 4y - 6 = 0\)
2. \(L_2: 12x - 5y - 3 = 0\)
3. \(L_3: 4x - 3y + 12 = 0\)

### Step 2: Determine the coefficients of the lines
For the lines \(L_2\) and \(L_3\), we will identify the coefficients:
- For \(L_2: 12x - 5y - 3 = 0\), we have:
- \(a_1 = 12\), \(b_1 = -5\), \(c_1 = -3\)

- For \(L_3: 4x - 3y + 12 = 0\), we have:
- \(a_2 = 4\), \(b_2 = -3\), \(c_2 = 12\)

### Step 3: Check the condition for angle bisector
We need to check if the angle bisector can be formed using these two lines. We calculate \(a_1a_2 + b_1b_2\):
\[ 12 \cdot 4 + (-5) \cdot (-3) = 48 + 15 = 63 > 0 \]
Since this is greater than zero, we can proceed to find the angle bisector.

### Step 4: Use the angle bisector formula
The formula for the internal angle bisector is given by:
\[ \frac{a_1 x + b_1 y + c_1}{\sqrt{a_1^2 + b_1^2}} = \frac{a_2 x + b_2 y + c_2}{\sqrt{a_2^2 + b_2^2}} \]

Substituting the values:
\[ \frac{12x - 5y - 3}{\sqrt{12^2 + (-5)^2}} = \frac{4x - 3y + 12}{\sqrt{4^2 + (-3)^2}} \]

Calculating the denominators:
\[ \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
\[ \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

Thus, we can rewrite the equation as:
\[ \frac{12x - 5y - 3}{13} = \frac{4x - 3y + 12}{5} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 5(12x - 5y - 3) = 13(4x - 3y + 12) \]

Expanding both sides:
\[ 60x - 25y - 15 = 52x - 39y + 156 \]

### Step 6: Rearranging the equation
Rearranging the equation to one side:
\[ 60x - 52x - 25y + 39y - 15 - 156 = 0 \]
\[ 8x + 14y - 171 = 0 \]

### Step 7: Final equation of the internal bisector
Thus, the equation of the internal bisector of the angle opposite to the side \(3x + 4y - 6 = 0\) is:
\[ 8x + 14y - 171 = 0 \]