What will be the angle between the lines ` y - x - 7 `and ` sqrt (3 ) y - x + 6 = 0 ` ?

Options

A. `theta = tan^(-1) ( 2 + sqrt(3))`
B. `theta = tan^(-1) ( 2 - sqrt(3))`
C. `theta = tan^(-1) ( 1 + sqrt(3))`
D. `theta = tan^(-1) ( 1 - sqrt(3))`

Answer

To find the angle between the lines given by the equations \( y - x - 7 = 0 \) and \( \sqrt{3}y - x + 6 = 0 \), we will follow these steps:

### Step 1: Rewrite the equations in slope-intercept form
The slope-intercept form of a line is given by \( y = mx + c \), where \( m \) is the slope.

1. For the first line \( y - x - 7 = 0 \):
\[ y = x + 7 \]
Here, the slope \( m_1 = 1 \).

2. For the second line \( \sqrt{3}y - x + 6 = 0 \):
\[ \sqrt{3}y = x - 6 \implies y = \frac{1}{\sqrt{3}}x - \frac{6}{\sqrt{3}} \]
Here, the slope \( m_2 = \frac{1}{\sqrt{3}} \).

### Step 2: Use the formula for the angle between two lines
The formula for the tangent of the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]

### Step 3: Substitute the slopes into the formula
Substituting \( m_1 = 1 \) and \( m_2 = \frac{1}{\sqrt{3}} \):
\[ \tan \theta = \left| \frac{\frac{1}{\sqrt{3}} - 1}{1 + 1 \cdot \frac{1}{\sqrt{3}}} \right| \]

### Step 4: Simplify the expression
1. The numerator:
\[ \frac{1}{\sqrt{3}} - 1 = \frac{1 - \sqrt{3}}{\sqrt{3}} \]

2. The denominator:
\[ 1 + \frac{1}{\sqrt{3}} = \frac{\sqrt{3} + 1}{\sqrt{3}} \]

Thus, we have:
\[ \tan \theta = \left| \frac{\frac{1 - \sqrt{3}}{\sqrt{3}}}{\frac{\sqrt{3} + 1}{\sqrt{3}}} \right| = \left| \frac{1 - \sqrt{3}}{\sqrt{3} + 1} \right| \]

### Step 5: Rationalize the expression
To rationalize:
\[ \tan \theta = \left| \frac{(1 - \sqrt{3})(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} \right| = \left| \frac{(1 - \sqrt{3})(\sqrt{3} - 1)}{3 - 1} \right| = \left| \frac{(1 - \sqrt{3})(\sqrt{3} - 1)}{2} \right| \]

Expanding the numerator:
\[ (1 - \sqrt{3})(\sqrt{3} - 1) = 1\sqrt{3} - 1 - \sqrt{3} + 3 = 4 - 2\sqrt{3} \]
Thus:
\[ \tan \theta = \left| \frac{4 - 2\sqrt{3}}{2} \right| = |2 - \sqrt{3}| \]

### Step 6: Final angle calculation
Since \( \tan \theta = |2 - \sqrt{3}| \), we can express the angle \( \theta \) as:
\[ \theta = \tan^{-1}(2 - \sqrt{3}) \]

### Conclusion
The angle between the two lines is given by:
\[ \theta = \tan^{-1}(2 - \sqrt{3}) \]