Question

Find the value of \( k \), if \( 2x + y = 0 \) is one of the lines represented by \( 3x^2 + kxy + 2y^2 = 0 \)

Hint:

For a pair of lines, substitute one line's equation into the quadratic form to solve for the coefficient.

Answer 1

The equation \( 3x^2 + kxy + 2y^2 = 0 \) represents a pair of lines. Let one line be \( 2x + y = 0 \), i.e., \( y = -2x \). 
Substitute \( y = -2x \) into \( 3x^2 + kxy + 2y^2 = 0 \): 
\[ 3x^2 + kx(-2x) + 2(-2x)^2 = 0 \Rightarrow 3x^2 - 2kx^2 + 2 \cdot 4x^2 = 0 \Rightarrow (3 - 2k + 8)x^2 = (11 - 2k)x^2 = 0. \] Since \( x^2 \neq 0 \), 
\[ 11 - 2k = 0 \Rightarrow 2k = 11 \Rightarrow k = \frac{11}{2}. \] Answer: \( k = \frac{11}{2} \).