If the slope of one of the lines given by $\mathrm{K} x^2+6 x y+y^2=0$ is three times the order, then the…

If the slope of one of the lines given by $\mathrm{K} x^2+6 x y+y^2=0$ is three times the order, then the value of K is
  1. $\frac{9}{4}$
  2. $\frac{4}{9}$
  3. $\frac{27}{4}$
  4. $\frac{4}{27}$

Solution

- Given equation of pair of lines is $\begin{array}{ll} & K x^2+6 x y+y^2=0 \\ \therefore \quad & A=K, H=3, B=1 \end{array}$
Let the slopes of the lines be $m_1$ and $m_2$ $m_1+m_2=\frac{-2 H}{B} \text { and } m_1 m_2=\frac{A}{B}$
Given that $\mathrm{m}_2=3 \mathrm{~m}_1$ $\therefore \quad m_1+3 m_1=\frac{-2 H}{B}=-6 \Rightarrow m_1=\frac{-3}{2}$ and $m_1 \times 3 m_1=\frac{A}{B}=K \Rightarrow 3 m_1^2=K \Rightarrow K=\frac{27}{4}$

Asked in: MHT CET 2024 (11 May Shift 1)

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