Find the value of $K$ . If the slope of one of the lines given by $4 x^{2} + K x y + y^{2} = 0$ is four times the other.

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Solution

The two lines are of form $y = m x$ and $y = 4 m x$
$∴$ combined equation is $\Rightarrow$
$(4 m x - y) (m x - y) = 0$
$\Rightarrow 4 m^{2} x^{2} + y^{2} - 5 m x y = 0$ ______ (1)
compare equation (1) with $4 x^{2} + k x y + y^{2}$
$4 m^{2} = 4$ and $K = - 5 m$
$∴ m = \pm 1 \Rightarrow K = \pm 5$

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