Question

What is the angle between the two straight lines \( y = (2 - \sqrt{3})x + 5 \) and \( y = (2 + \sqrt{3})x - 7 \)?

• A. \( 60^\circ \)
• B. \( 45^\circ \)
• C. \( 30^\circ \)
• D. \( 15^\circ \)

Hint:

The angle between two lines can be found using the formula \( \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \). If \( \tan \theta = \sqrt{3} \), then \( \theta = 60^\circ \).

Answer 1

The Correct Option is A

The general equation of a straight line is: \[ y = mx + c \] where \( m \) is the slope of the line. Comparing with the given equations: 
- For \( y = (2 - \sqrt{3})x + 5 \), the slope \( m_1 = 2 - \sqrt{3} \). 
- For \( y = (2 + \sqrt{3})x - 7 \), the slope \( m_2 = 2 + \sqrt{3} \). The formula for the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is: \[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \] 
Step 1: Substituting values \[ \tan \theta = \left| \frac{(2 + \sqrt{3}) - (2 - \sqrt{3})}{1 + (2 - \sqrt{3})(2 + \sqrt{3})} \right| \] Simplifying the numerator: \[ (2 + \sqrt{3}) - (2 - \sqrt{3}) = 2 + \sqrt{3} - 2 + \sqrt{3} = 2\sqrt{3} \] 
Simplifying the denominator: \[ 1 + (2 - \sqrt{3})(2 + \sqrt{3}) \] Using the identity \( (a - b)(a + b) = a^2 - b^2 \): \[ 1 + [4 - 3] = 1 + 1 = 2 \] 
Thus, \[ \tan \theta = \left| \frac{2\sqrt{3}}{2} \right| = \left| \sqrt{3} \right| \] \[ \tan \theta = \sqrt{3} \] Since \( \tan 60^\circ = \sqrt{3} \), we get: \[ \theta = 60^\circ \] Thus, the angle between the given lines is \( 60^\circ \).