If the slope of one of the lines in the pair of lines $8 x^2+a x y+y^2=0$ is thrice the slope of the second…

If the slope of one of the lines in the pair of lines $8 x^2+a x y+y^2=0$ is thrice the slope of the second line, then $a=$
  1. $8 \sqrt{\frac{2}{3}}$
  2. $6$
  3. $16 \sqrt{2}$
  4. $3 \frac{\sqrt{2}}{5}$

Solution

$8 x^2+a x y+y^2=0$
Let $m_1$ and $m_2$ be two slopes Since given equation is homogeneous $\Rightarrow m_1+m_2=\frac{-a}{1}$ and $m_1 m_2=8 ;$ Since $m_1=3 m_2$ $\therefore 3 m_2+m_2=-\mathrm{a} \Rightarrow 4 m_2=-\mathrm{a}$ ...(i) and $3 m_2^2=8 \Rightarrow m_2= \pm \sqrt{\frac{8}{3}}$ So, from (i), $-4 \sqrt{\frac{8}{3}}=-a \Rightarrow a=8 \sqrt{2 / 3}$

Asked in: AP EAMCET 2024 (22 May Shift 2)

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