Question

If one of the lines given by the equation \( x^2 + kxy + 2y^2 = 0 \) is \( x + 2y = 0 \), then the value of \( k \) is

• A. \( 2 \)
• B. \( 1 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

To find parameters in pair-of-lines equations, substitute the given line equation directly.

Answer 1

The Correct Option is C

Step 1: Use the condition for a pair of lines through the origin.
Given equation represents a pair of straight lines through the origin. If \( x + 2y = 0 \) is one line, then substituting it must satisfy the equation.

Step 2: Substitute \( x = -2y \).
\[ (-2y)^2 + k(-2y)(y) + 2y^2 = 0 \] \[ 4y^2 - 2ky^2 + 2y^2 = 0 \]

Step 3: Simplify.
\[ (6 - 2k)y^2 = 0 \Rightarrow 6 - 2k = 0 \] \[ k = 3 \]

Step 4: Conclusion.
\[ \boxed{k = 3} \]