Find the value of $k$ , if $2 x + y = 0$ is one of the lines given by ${3 x}^{2} + k x y + 2 y^{2} = 0$

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Solution

$2 x + y = 0$
$2 x = - y$
$3 x^{2} + k x y + 2 y^{2} = 0$
$3 x^{2} + k x (- 2 x) + 2 (4 x^{2}) = 0$
$3 x^{2} - 2 k x^{2} + 8 x^{2} = 0$
$k = \frac{11}{2}$ .

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