Pair of lines through `(1, 1)` and making equal angle with `3x - 4y=1` and `12x +9y=1` intersect x-axis at `P_1` and `P_2` , then `P_1,P_2` may be

Options

A. (a) `((8)/(7),0)` and `((9)/(7),0)`
B. (b) `((6)/(7),0)` and `(8,0)`
C. (c) `((8)/(7),0)` and `((1)/(8),0)`
D. (d) `(8,0)` and `((1)/(8),0)`

Explanation

To solve the problem of finding the points \( P_1 \) and \( P_2 \) where the pair of lines through the point \( (1, 1) \) intersects the x-axis, making equal angles with the lines \( 3x - 4y = 1 \) and \( 12x + 9y = 1 \), we can follow these steps:

### Step 1: Find the slopes of the given lines

1. **Convert the equations to slope-intercept form (y = mx + b)**:
- For the line \( 3x - 4y = 1 \):
\[ 4y = 3x - 1 \implies y = \frac{3}{4}x - \frac{1}{4} \]
Thus, the slope \( m_1 = \frac{3}{4} \).

- For the line \( 12x + 9y = 1 \):
\[ 9y = -12x + 1 \implies y = -\frac{12}{9}x + \frac{1}{9} \implies y = -\frac{4}{3}x + \frac{1}{9} \]
Thus, the slope \( m_2 = -\frac{4}{3} \).

### Step 2: Use the angle bisector property

Since the lines make equal angles with the new lines, we can use the formula for the slopes of the angle bisectors:
\[ \frac{m - m_1}{1 + m m_1} = \pm \frac{m - m_2}{1 + m m_2} \]

### Step 3: Set up the equations

1. **Using the positive case**:
\[ \frac{m - \frac{3}{4}}{1 + m \cdot \frac{3}{4}} = \frac{m + \frac{4}{3}}{1 - m \cdot \frac{4}{3}} \]

2. **Cross-multiplying and simplifying**:
- This leads to a quadratic equation in terms of \( m \).

3. **Using the negative case**:
\[ \frac{m - \frac{3}{4}}{1 + m \cdot \frac{3}{4}} = -\frac{m + \frac{4}{3}}{1 - m \cdot \frac{4}{3}} \]

4. **Cross-multiplying and simplifying**:
- This will also lead to a quadratic equation.

### Step 4: Solve for \( m \)

After solving the equations from both cases, we will find two possible slopes \( m_1 \) and \( m_2 \).

### Step 5: Write the equations of the lines

Using the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \):
1. For \( m_1 \):
\[ y - 1 = m_1(x - 1) \]

2. For \( m_2 \):
\[ y - 1 = m_2(x - 1) \]

### Step 6: Find the intersection points with the x-axis

1. Set \( y = 0 \) in both line equations to find \( P_1 \) and \( P_2 \):
- For the first line, solve for \( x \) when \( y = 0 \).
- For the second line, similarly solve for \( x \) when \( y = 0 \).

### Step 7: Identify the points \( P_1 \) and \( P_2 \)

After solving, we will find the coordinates of \( P_1 \) and \( P_2 \).

### Final Answer

The points \( P_1 \) and \( P_2 \) are:
- \( P_1 = (8, 0) \)
- \( P_2 = \left(\frac{6}{7}, 0\right) \)