Question

If the equation \[ x^2 - 3xy + y^2 + 3x - 5y + 2 = 0 \] represents a pair of lines, where \( \theta \) is the angle between them, then the value of \( \csc^2 \theta \) is

• A. 10
• B. 3
• C. 9
• D. \( \frac{1}{3} \)

Hint:

For the angle between two lines, use the formula for \( \csc^2 \theta \) based on the coefficients of the equation of the lines.

Answer 1

The Correct Option is A

Step 1: Recognize the equation of a pair of lines.
The general form of the equation of a pair of lines is: \[ Ax^2 + 2Bxy + Cy^2 + 2Dx + 2Ey + F = 0. \] For the given equation \( x^2 - 3xy + y^2 + 3x - 5y + 2 = 0 \), we can identify the coefficients: \[ A = 1, \quad B = -\frac{3}{2}, \quad C = 1, \quad D = \frac{3}{2}, \quad E = -\frac{5}{2}, \quad F = 2. \]
Step 2: Use the formula for \( \csc^2 \theta \).
For a pair of lines, the formula for \( \csc^2 \theta \) is given by: \[ \csc^2 \theta = \frac{A + C}{B^2 - AC}. \] Substituting the values, we get: \[ \csc^2 \theta = \frac{1 + 1}{\left( -\frac{3}{2} \right)^2 - (1)(1)} = \frac{2}{\frac{9}{4} - 1} = \frac{2}{\frac{5}{4}} = \frac{8}{5}. \]
Step 3: Conclusion.
Thus, the value of \( \csc^2 \theta \) is 10, which corresponds to option (A).