Suppose the slopes $m_1$ and $m_2$ of the lines represented by $a x^2+2 h x y+b y^2=0$ satisfy $3\left(m_1-m_2\right)-7=0$ and $m_1 m_2-2=0$. Then, which of the following is true?

• Correct answer
  A

  $\frac{a}{12}=\frac{b}{6}=\frac{h}{ \pm 11}$
• B

  $\frac{a}{6}=\frac{b}{12}=\frac{h}{ \pm 11}$
• C

  $a=b= \pm h$
• D

  $\frac{a}{2}=b= \pm h$

Solution

$y-m_1 x=0$ ...(i) $y-m_2 x=0$ ...(ii) $\Rightarrow\left(y-m_1 x\right)\left(y-m_2 x\right)=0$ $y^2-\left(m_1+m_2\right) x y+m_1 m_2 x^2=0$ Now, $a x^2+2 h x y+b y^2=0$ $\left(\frac{a}{b}\right) x^2+\frac{2 h}{b} x y+y^2=0 \Rightarrow m_1 \cdot m_2=\frac{a}{b}$ $\therefore \quad 2=\frac{a}{b} \Rightarrow a=2 b$ $\Rightarrow \quad \frac{a}{12}=\frac{b}{6}$

Hence, the correct answer is option: 'A'.