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…

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
  1. 4
  2. 2.5
  3. 5
  4. 1

Solution

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), $\mathrm{k}= \pm 5$

Asked in: MHT CET 2024 (15 May Shift 1)

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