Find k if one of the line is given by
$6 x^{2} + k x y + y^{2} = 0 i s 2 x + y = 0$

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Solution

$6 x^{2} + k x y + y^{2} = 0$ $\Rightarrow a x^{2} + 2 h x y + b y^{2} = 0$
$2 x + y = 0$ $a = 6 h = \frac{k}{2} b = 1$
$m_{1} + m_{2} = \frac{- 2 h}{b} = - k$
$- 2 + m_{2} = - k$
$m_{2} = - k + 2$
$m_{1} m_{2} = a / b$

$- 2 (- k + 2) = 6$
$2 k - 4 = 6$
$2 k = 10$
$k = 5$

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