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

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Solution

Let $m_{1}$ be the slope of $2 x + y = 0$
$∴ m_{1} = - 2$
Now, comparing $6 x^{2} + k x y + y^{2} = 0$ with $a x^{2} + 2 h x y + b y^{2} = 0$ , we get
$a = 6, h = \frac{k}{2}, b = 1$
$∴ m_{1} + m_{2} = \frac{- 2 h}{b} = - k$
$∴ - 2 + m_{2} = - k \Rightarrow m_{2} = 2 - k$
Now, $m_{1} \cdot m_{2} = \frac{a}{b}$
$\Rightarrow (- 2) (2 - k) = \frac{6}{1}$
$\Rightarrow - 4 + 2 k = 6$
$\Rightarrow k = 5$

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