If $2 x + y = 0$ is one of the lines represented by $3 x^{2} + k x y + 2 y^{2} = 0,$ then find the value of $k$

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Solution

Auxiliary equation of the given equation is $2 m^{2} + k m + 3 = 0$

Slope of the line $2 x + y = 0$ is $m = - 2$

$∴ m = - 2$ is a root of the auxilliary equation $2 m^{2} + k m + 3 = 0$

$∴ 2 {(- 2)}^{2} - 2 k + 3 = 0$

$∴ 8 - 2 k + 3 = 0$

$∴ - 2 k + 11 = 0$

$\Rightarrow - 2 k = - 11$

$∴ k = \frac{11}{2}$

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