Find the angles between the pairs of straight lines
`x- 4y = 3 and 6x- y = 11`

Answer

To find the angle between the two straight lines given by the equations \( x - 4y = 3 \) and \( 6x - y = 11 \), we will follow these steps:

### Step 1: Convert the equations to slope-intercept form
We need to rearrange both equations into the form \( y = mx + c \), where \( m \) is the slope.

**For the first line:**
1. Start with the equation:
\[ x - 4y = 3 \]
2. Rearranging gives:
\[ -4y = -x + 3 \]
3. Dividing by -4:
\[ y = \frac{1}{4}x - \frac{3}{4} \]
Here, the slope \( m_1 = \frac{1}{4} \).

**For the second line:**
1. Start with the equation:
\[ 6x - y = 11 \]
2. Rearranging gives:
\[ -y = -6x + 11 \]
3. Dividing by -1:
\[ y = 6x - 11 \]
Here, the slope \( m_2 = 6 \).

### Step 2: Use the formula for 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 slopes into the formula
Substituting \( m_1 = \frac{1}{4} \) and \( m_2 = 6 \):
\[ \tan \theta = \left| \frac{\frac{1}{4} - 6}{1 + \frac{1}{4} \cdot 6} \right| \]

### Step 4: Simplify the expression
1. Calculate \( m_1 - m_2 \):
\[ \frac{1}{4} - 6 = \frac{1}{4} - \frac{24}{4} = \frac{-23}{4} \]
2. Calculate \( 1 + m_1 m_2 \):
\[ 1 + \frac{1}{4} \cdot 6 = 1 + \frac{6}{4} = 1 + \frac{3}{2} = \frac{5}{2} \]
3. Now substitute these values back into the formula:
\[ \tan \theta = \left| \frac{\frac{-23}{4}}{\frac{5}{2}} \right| = \left| \frac{-23}{4} \cdot \frac{2}{5} \right| = \left| \frac{-46}{20} \right| = \frac{46}{20} = \frac{23}{10} \]

### Step 5: Find the angle \( \theta \)
To find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}\left(\frac{23}{10}\right) \]

### Conclusion
The angle between the two lines \( x - 4y = 3 \) and \( 6x - y = 11 \) is given by:
\[ \theta = \tan^{-1}\left(\frac{23}{10}\right) \]