The equation of the anlge bisector of the angle formed by the lines `12x+5y-10=0` and 3x + 4y - 5 = 0 containing the point (1,1) is ………………

Answer

To find the equation of the angle bisector of the lines given by the equations \(12x + 5y - 10 = 0\) and \(3x + 4y - 5 = 0\) that contains the point \((1, 1)\), we can follow these steps:

### Step 1: Identify the coefficients of the lines
For the first line \(12x + 5y - 10 = 0\):
- \(a_1 = 12\)
- \(b_1 = 5\)
- \(c_1 = -10\)

For the second line \(3x + 4y - 5 = 0\):
- \(a_2 = 3\)
- \(b_2 = 4\)
- \(c_2 = -5\)

### Step 2: Write the formula for the angle bisector
The equation of the angle bisector can be written as:
\[ \frac{a_1 x + b_1 y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2 x + b_2 y + c_2}{\sqrt{a_2^2 + b_2^2}} \]

### Step 3: Calculate the denominators
Calculate \(\sqrt{a_1^2 + b_1^2}\) and \(\sqrt{a_2^2 + b_2^2}\):
\[ \sqrt{a_1^2 + b_1^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
\[ \sqrt{a_2^2 + b_2^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Substitute into the angle bisector equation
Substituting the values into the angle bisector equation gives:
\[ \frac{12x + 5y - 10}{13} = \pm \frac{3x + 4y - 5}{5} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 5(12x + 5y - 10) = 13(3x + 4y - 5) \]

### Step 6: Expand both sides
Expanding both sides:
\[ 60x + 25y - 50 = 39x + 52y - 65 \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ 60x - 39x + 25y - 52y = -65 + 50 \]
\[ 21x - 27y = -15 \]

### Step 8: Simplify the equation
Dividing the entire equation by 3:
\[ 7x - 9y = -5 \]

### Step 9: Write the final equation
Rearranging gives the final equation of the angle bisector:
\[ 7x - 9y + 5 = 0 \]

### Step 10: Verify which bisector contains the point (1, 1)
To determine which angle bisector contains the point \((1, 1)\), we can substitute \(x = 1\) and \(y = 1\) into both forms of the angle bisector equation:
1. For the positive case:
\[ 5(12(1) + 5(1) - 10) = 13(3(1) + 4(1) - 5) \]
This simplifies to \(5(7) = 13(2)\) which is true.

2. For the negative case:
\[ 5(12(1) + 5(1) - 10) = -13(3(1) + 4(1) - 5) \]
This simplifies to \(5(7) = -13(2)\) which is false.

Since the point \((1, 1)\) satisfies the positive case, the angle bisector containing the point \((1, 1)\) is:
\[ 7x - 9y + 5 = 0 \]