Question:
The value of the angle between two straight lines is \(y = (2 - \sqrt{3})x + 5\) and \(y = (2 + \sqrt{3})x - 7\) is
The value of the angle between two straight lines is \(y = (2 - \sqrt{3})x + 5\) and \(y = (2 + \sqrt{3})x - 7\) is
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Use \(\tan \theta\) formula for angle between two lines using slopes.
Updated On: Apr 16, 2026
- \(30^\circ\)
- \(60^\circ\)
- \(45^\circ\)
- \(90^\circ\)
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The Correct Option is B
Solution and Explanation
Concept:
Angle between two lines with slopes \(m_1\) and \(m_2\):
\[
\tan \theta = \left|\frac{m_2 - m_1}{1 + m_1 m_2}\right|
\]
Step 1: Identify slopes.
\[ m_1 = 2 - \sqrt{3}, \quad m_2 = 2 + \sqrt{3} \]
Step 2: Compute numerator.
\[ m_2 - m_1 = (2+\sqrt{3}) - (2-\sqrt{3}) = 2\sqrt{3} \]
Step 3: Compute denominator.
\[ 1 + m_1 m_2 = 1 + (2-\sqrt{3})(2+\sqrt{3}) \] \[ = 1 + (4 - 3) = 2 \]
Step 4: Find angle.
\[ \tan \theta = \frac{2\sqrt{3}}{2} = \sqrt{3} \Rightarrow \theta = 60^\circ \]
Step 1: Identify slopes.
\[ m_1 = 2 - \sqrt{3}, \quad m_2 = 2 + \sqrt{3} \]
Step 2: Compute numerator.
\[ m_2 - m_1 = (2+\sqrt{3}) - (2-\sqrt{3}) = 2\sqrt{3} \]
Step 3: Compute denominator.
\[ 1 + m_1 m_2 = 1 + (2-\sqrt{3})(2+\sqrt{3}) \] \[ = 1 + (4 - 3) = 2 \]
Step 4: Find angle.
\[ \tan \theta = \frac{2\sqrt{3}}{2} = \sqrt{3} \Rightarrow \theta = 60^\circ \]
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