find the value of k , the slope of one of the the line given by 4x^​2 +kxy + y² = 0 is four times the other .
 

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{for} {\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2=0 \\ {\mathrm{m}}_1+{\mathrm{m}}_2=-\frac{2\mathrm{h}}{\mathrm{b}} \\ {\mathrm{m}}_1{\mathrm{m}}_2=\frac{\mathrm{a}}{\mathrm{b}} \\ \mathrm{Here} \mathrm{one} \mathrm{slope} \mathrm{is} 4 \mathrm{times} \mathrm{the} \mathrm{other}, \mathrm{hence} \mathrm{let} \mathrm{slopes} \mathrm{be} \mathrm{m} \mathrm{and} 4\mathrm{m}. \\ 4{\mathrm{x}}^2+\mathrm{kxy}+{\mathrm{y}}^2=0 \\ \mathrm{m}+4\mathrm{m}=-\frac{\mathrm{k}}{1} \\ \Rightarrow \mathrm{m}=-\frac{\mathrm{k}}{5} \\ \mathrm{m}.4\mathrm{m}=\frac{4}{1} \\ \Rightarrow {\mathrm{m}}^2=1 \\ \Rightarrow \frac{{\mathrm{k}}^2}{25}=1 \\ \Rightarrow {\mathrm{k}}^2=25 \\ \Rightarrow \mathrm{k}=\pm 5 \left(\mathrm{Answer}\right) \end{gathered}$$

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