Question

If the angle between the lines given by the equation \[ x^2 - 3xy + Ay^2 + 3x - 5y + 2 = 0, \quad \lambda \geq 0, \quad \text{is} \, \tan^{-1} \left( \frac{1}{3} \right), \, \text{then} \, \lambda = \]

• A. \( \frac{2}{3}, 40 \)
• B. 10
• C. \( \frac{2}{5}, 1 \)
• D. 2

Hint:

To calculate the angle between two lines from their equation, use the formula involving the slopes and the condition given in the problem.

Answer 1

The Correct Option is D

Step 1: Using the formula for the angle between two lines.
The general equation for the angle between two lines is given by: \[ \tan \theta = \frac{|m_1 - m_2|}{1 + m_1 m_2} \] where \( m_1 \) and \( m_2 \) are the slopes of the lines. For the given quadratic equation, we use the condition for the angle between the lines and the discriminant method to find \( \lambda \).

Step 2: Finding the value of \( \lambda \).
After solving for \( \lambda \), we get \( \lambda = 2 \).

Step 3: Conclusion.
Thus, the value of \( \lambda \) is 2, which makes option (D) the correct answer.