Question

The equation \( x^2 - 3xy + \lambda y^2 + 3x - 5y + 2 = 0 \), where \( \lambda \) is a real number represents a pair of lines. If \( \theta \) is the acute angle between the lines, then \( \frac{\text{cosec}^2 \theta}{\sqrt{10}} = \)}

• A. 10
• B. \( \frac{1}{\sqrt{10}} \)
• C. 2
• D. \( \sqrt{10} \)

Hint:

The angle between lines depends only on the second-degree terms ($ax^2 + 2hxy + by^2$).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The general second-degree equation represents a pair of lines if its discriminant \( \Delta = 0 \). We first find \( \lambda \), then the angle between the lines.
Step 2: Key Formula or Approach:
For \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \):
1. \( \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \)
2. \( \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a+b} \right| \)
Step 3: Detailed Explanation:
Comparing with the general equation: \( a=1, 2h=-3 \implies h=-1.5, b=\lambda, 2g=3 \implies g=1.5, 2f=-5 \implies f=-2.5, c=2 \).
Using \( \Delta = 0 \):
\[ 1(\lambda)(2) + 2(-2.5)(1.5)(-1.5) - 1(-2.5)^2 - \lambda(1.5)^2 - 2(-1.5)^2 = 0 \] \[ 2\lambda + 11.25 - 6.25 - 2.25\lambda - 4.5 = 0 \] \[ -0.25\lambda + 0.5 = 0 \implies \lambda = 2 \] Now calculate \( \tan \theta \):
\[ \tan \theta = \frac{2\sqrt{(-1.5)^2 - (1)(2)}}{1+2} = \frac{2\sqrt{2.25 - 2}}{3} = \frac{2\sqrt{0.25}}{3} = \frac{2 \times 0.5}{3} = \frac{1}{3} \] Since \( \tan \theta = \frac{1}{3} \), we have \( \text{cot } \theta = 3 \).
\( \text{cosec}^2 \theta = 1 + \text{cot}^2 \theta = 1 + 9 = 10 \).
The required value is \( \frac{10}{\sqrt{10}} = \sqrt{10} \).
Step 4: Final Answer:
The value is \( \sqrt{10} \).