Question

If the angles between the pair of straight lines represented by the equation \(x^2 - 3xy + \lambda y^2 + 3x - 5y + 2 = 0\) is \(\tan^{-1}(1/3)\). Where \(\lambda\) is a non-negative real number, then \(\lambda\) is

• A. 2
• B. 0
• C. 3
• D. 1

Hint:

Angle between lines: \(\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For pair of lines \(ax^2 + 2hxy + by^2 = 0\), \(\tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b}\).
Step 2: Detailed Explanation:
Here \(a = 1\), \(2h = -3\), \(h = -3/2\), \(b = \lambda\)
\(\tan\theta = 1/3 = \frac{2\sqrt{9/4 - \lambda}}{1 + \lambda}\)
\((1 + \lambda) = 3\sqrt{9 - 4\lambda}\)
Square: \(1 + \lambda^2 + 2\lambda = 81 - 36\lambda\)
\(\lambda^2 + 38\lambda - 80 = 0 \rightarrow (\lambda - 2)(\lambda + 40) = 0\)
\(\lambda = 2\) (non-negative)
Step 3: Final Answer:
\(\lambda = 2\).