Question

The acute angle between the lines given by \( y - \sqrt{3}x + 1 = 0 \) and \( \sqrt{3}y - x + 7 = 0 \) is

• A. \( 75^\circ \)
• B. \( 60^\circ \)
• C. \( 45^\circ \)
• D. \( 30^\circ \)

Hint:

To find the angle between two lines, use the formula for the tangent of the angle based on the slopes of the lines.

Answer 1

The Correct Option is D

Step 1: Find the slopes of the two lines.
The equation of the first line is \( y = \sqrt{3}x - 1 \), so the slope is \( m_1 = \sqrt{3} \). The equation of the second line is \( y = \frac{1}{\sqrt{3}}x - \frac{7}{\sqrt{3}} \), so the slope is \( m_2 = \frac{1}{\sqrt{3}} \).
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: \[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \] Substitute \( m_1 = \sqrt{3} \) and \( m_2 = \frac{1}{\sqrt{3}} \) into the formula: \[ \tan \theta = \left| \frac{\sqrt{3} - \frac{1}{\sqrt{3}}}{1 + \sqrt{3} \cdot \frac{1}{\sqrt{3}}} \right| = \left| \frac{\frac{2\sqrt{3}}{3}}{2} \right| = \frac{1}{\sqrt{3}} \]
Step 3: Conclusion.
Thus, the acute angle between the lines is \( 30^\circ \).