Find the angle between two lines `x+2y-111=0` and `x-3y-19=0`

Answer

To find the angle between the two lines given by the equations \( x + 2y - 111 = 0 \) and \( x - 3y - 19 = 0 \), we can follow these steps:

### Step 1: Convert the equations to slope-intercept form (y = mx + c)

1. **For the first line** \( x + 2y - 111 = 0 \):
\[ 2y = -x + 111 \]
\[ y = -\frac{1}{2}x + \frac{111}{2} \]
Here, the slope \( m_1 = -\frac{1}{2} \).

2. **For the second line** \( x - 3y - 19 = 0 \):
\[ -3y = -x + 19 \]
\[ y = \frac{1}{3}x - \frac{19}{3} \]
Here, the slope \( m_2 = \frac{1}{3} \).

### Step 2: Use the formula to find the angle between two lines

The formula for the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

### Step 3: Substitute the values of \( m_1 \) and \( m_2 \)

Substituting \( m_1 = -\frac{1}{2} \) and \( m_2 = \frac{1}{3} \):
\[ \tan \theta = \left| \frac{-\frac{1}{2} - \frac{1}{3}}{1 + \left(-\frac{1}{2}\right) \left(\frac{1}{3}\right)} \right| \]

### Step 4: Simplify the expression

1. **Calculate the numerator**:
\[ -\frac{1}{2} - \frac{1}{3} = -\frac{3}{6} - \frac{2}{6} = -\frac{5}{6} \]

2. **Calculate the denominator**:
\[ 1 + \left(-\frac{1}{2}\right) \left(\frac{1}{3}\right) = 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6} \]

3. **Combine the results**:
\[ \tan \theta = \left| \frac{-\frac{5}{6}}{\frac{5}{6}} \right| = \left| -1 \right| = 1 \]

### Step 5: Find the angle \( \theta \)

Since \( \tan \theta = 1 \), we have:
\[ \theta = 45^\circ \]

### Final Answer:
The angle between the two lines is \( 45^\circ \).

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