The angle between the lines `y = (2-sqrt(3))X + 5 and y = (2+sqrt(3))X - 7` is

Options

A. `30^(@)`
B. `60^(@)`
C. `45^(@)`
D. `tan^(-1) 3`

Answer

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

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

For the first line:
\[ y = (2 - \sqrt{3})X + 5 \]
The slope \( m_1 = 2 - \sqrt{3} \).

For the second line:
\[ y = (2 + \sqrt{3})X - 7 \]
The slope \( m_2 = 2 + \sqrt{3} \).

### Step 2: Use the formula for the angle between two lines
The formula to find 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 \( m_2 \) into the formula:
\[ \tan \theta = \left| \frac{(2 - \sqrt{3}) - (2 + \sqrt{3})}{1 + (2 - \sqrt{3})(2 + \sqrt{3})} \right| \]

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

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

### Step 6: Substitute back into the formula
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
To find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}(\sqrt{3}) \]
From trigonometric values, we know that:
\[ \tan \frac{\pi}{3} = \sqrt{3} \]
Thus,
\[ \theta = \frac{\pi}{3} \text{ radians} = 60^\circ \]

### Final Answer
The angle between the lines is \( 60^\circ \).

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