Find the angle between the lines `y=(2-sqrt(3))(x+5)andy=(2+sqrt(3))(x-7)`

Answer

To find the angle between the two lines given by the equations \( y = (2 - \sqrt{3})(x + 5) \) and \( y = (2 + \sqrt{3})(x - 7) \), we can follow these steps:

### Step 1: Identify the slopes of the lines
The equations of the lines can be rewritten in slope-intercept form \( y = mx + b \), where \( m \) is the slope.

For the first line:
\[ y = (2 - \sqrt{3})(x + 5) \]
Expanding this:
\[ y = (2 - \sqrt{3})x + (2 - \sqrt{3}) \cdot 5 \]
Thus, the slope \( m_1 \) of the first line is:
\[ m_1 = 2 - \sqrt{3} \]

For the second line:
\[ y = (2 + \sqrt{3})(x - 7) \]
Expanding this:
\[ y = (2 + \sqrt{3})x - (2 + \sqrt{3}) \cdot 7 \]
Thus, the slope \( m_2 \) of the second line is:
\[ m_2 = 2 + \sqrt{3} \]

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

### Step 4: Simplify the numerator
Calculating the numerator:
\[ (2 - \sqrt{3}) - (2 + \sqrt{3}) = 2 - \sqrt{3} - 2 - \sqrt{3} = -2\sqrt{3} \]

### Step 5: Simplify the denominator
Calculating the denominator:
\[ 1 + (2 - \sqrt{3})(2 + \sqrt{3}) = 1 + (4 - 3) = 1 + 1 = 2 \]

### Step 6: Combine the results
Now substituting back into the formula:
\[ \tan \theta = \left| \frac{-2\sqrt{3}}{2} \right| = \left| -\sqrt{3} \right| = \sqrt{3} \]

### Step 7: Find the angle \( \theta \)
Since \( \tan \theta = \sqrt{3} \), we find \( \theta \):
\[ \theta = 60^\circ \]

Thus, the angle between the two lines is \( 60^\circ \).