The value of \(k\), if the slope of one of the lines given by \(4 x^2+\mathrm{k} x y+y^2=0\) is four times that of the other, is given by

Given equation of pair of lines is
\(\begin{aligned}
& 4 x^2+\mathrm{k} x y+y^2=0 \\
\therefore \quad & \mathrm{a}=4, \mathrm{~h}=\frac{\mathrm{k}}{2}, \mathrm{~b}=1
\end{aligned}\)
According to the given condition,
\(\begin{aligned}
& \mathrm{m}_1=4 \mathrm{~m}_2 \\
& \mathrm{~m}_1+\mathrm{m}_2=-\mathrm{k} \\
& \Rightarrow 4 \mathrm{~m}_2+\mathrm{m}_2=-\mathrm{k} \\
& \Rightarrow 5 \mathrm{~m}_2=-\mathrm{k} \\
& \Rightarrow \mathrm{~m}_2=\frac{\mathrm{k}}{5} ...(i)\\
& \mathrm{~m}_1 \mathrm{~m}_2=4 \\
& \Rightarrow\left(4 \mathrm{~m}_2\right) \mathrm{m}_2=4 \\
& \Rightarrow \mathrm{~m}_2^2=1 \\
& \Rightarrow \mathrm{~m}_2= \pm 1
\end{aligned}\)
From (i), \(\mathrm{k}= \pm 5\)