Question

The auxiliary equation of the lines passing through the origin and having slopes \[ \sqrt{3} + 1 \quad \text{and} \quad \sqrt{3} - 1 \quad \text{is} \]

• A. \( m^2 - 2\sqrt{3}m + 2 = 0 \)
• B. \( m^2 - 2\sqrt{3}m - 2 = 0 \)
• C. \( m^2 + 2\sqrt{3}m - 2 = 0 \)
• D. \( m^2 + 2\sqrt{3}m + 2 = 0 \)

Hint:

For lines passing through the origin with given slopes, use the auxiliary equation \( m^2 - (m_1 + m_2)m + m_1m_2 = 0 \) to find the relationship between the slopes.

Answer 1

The Correct Option is A

Step 1: Understand the equation for the lines.
For the lines passing through the origin, the auxiliary equation is given by: \[ m^2 - (m_1 + m_2) m + m_1 m_2 = 0, \] where \( m_1 \) and \( m_2 \) are the slopes of the lines.
Step 2: Apply the slopes.
Here, \( m_1 = \sqrt{3} + 1 \) and \( m_2 = \sqrt{3} - 1 \). Thus, the equation becomes: \[ m^2 - \left( (\sqrt{3} + 1) + (\sqrt{3} - 1) \right) m + (\sqrt{3} + 1)(\sqrt{3} - 1) = 0. \] Simplifying, we get: \[ m^2 - 2\sqrt{3} m + 2 = 0. \] Thus, the correct answer is option (A).