What is the angle between the two straight lines `y = ( 2 - sqrt3) x + 5` and y `= (2 + sqrt3) x - 7` ?
(a)`60^(@)`
(b)`45^(@)`
(c)`30^(@)`
(d)`15^(@)`

Options

A. `60^(@)`
B. `45^(@)`
C. `30^(@)`
D. `15^(@)`

Answer

To find the angle between the two straight 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 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 tangent of 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:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]

### Step 3: Substitute the slopes into the formula
Now, substituting \( m_1 \) and \( m_2 \) into the formula:
\[ \tan \theta = \frac{(2 - \sqrt{3}) - (2 + \sqrt{3})}{1 + (2 - \sqrt{3})(2 + \sqrt{3})} \]

### 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 we have:
\[ \tan \theta = \frac{-2\sqrt{3}}{2} = -\sqrt{3} \]

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

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

### Final Answer
The correct option is (a) \( 60^\circ \).

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