The value of k,if the slope of one of the lin | vedclass

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(C) The given equation of the pair of lines is $4x^2 + kxy + y^2 = 0$. Comparing this with the general form $ax^2 + 2hxy + by^2 = 0$,we get $a = 4$,$2

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$5$

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(C) The given equation of the pair of lines is $4x^2 + kxy + y^2 = 0$. Comparing this with the general form $ax^2 + 2hxy + by^2 = 0$,we get $a = 4$,$2h = k$ (so $h = k/2$),and $b = 1$. Let the slopes of the two lines be $m_1$ and $m_2$. We know that $m_1 + m_2 = -\frac{2h}{b} = -k$ and $m_1 m_2 = \frac{a}{b} = 4$. According to the given condition,$m_1 = 4m_2$. Substituting this into the sum of slopes: $4m_2 + m_2 = -k \implies 5m_2 = -k \implies m_2 = -k/5$. Substituting into the product of slopes: $(4m_2)(m_2) = 4 \implies 4m_2^2 = 4 \implies m_2^2 = 1 \implies m_2 = \pm 1$. Since $m_2 = -k/5$,we have $-k/5 = \pm 1$,which gives $k = \mp 5$. Considering the magnitude or the standard form of such problems,the value is $k = \pm 5$. Given the options,$5$ is the correct magnitude.

The value of $k$,if the slope of one of the lines given by $4x^2 + kxy + y^2 = 0$ is four times that of the other,is given by

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