Question

I: lf the equation $4x^2+mxy-3y^2=0$ represents a pair of real and distinct lines, then $m\in R.$
II: lf the difference of the slopes of the lines $x^2-12kxy+y^2=0$ is $2,$ then $k$ is $\frac{1}{30}$.

• A. only I is true
• B. Only II is true
• C. both I and II are true
• D. neither I nor II are true

Answer 1

The correct option is A only I is true

1. $4x^2+mxy-3y^2=0$
$t=\frac{x}{y},$ $4t^2+tm-3=0$
$t=\frac{-m\pm \sqrt{m^2+48}}{8}$
for real lines, $m^2+48>0$
$\Rightarrow m\in R$
2. $y=m_{1}x$ and $y=m_{2}x$
So, joint equation is $y^2-(m_1+m_2)xy+m_1{m}_2{x}^2=0$
$\therefore x^2-\left(\frac{m_1+m_2}{m_1{m}_2}\right)xy+\frac{1}{m_1{m}_2}{y}^2=0$
Given, $x^2-12xy+y^2=0$
$\therefore \frac{m_1+m_2}{m_1{m}_2}=12$ ----(1)
& $m_1{m}_2=1$
$\therefore m_1+m_2=12$
$\therefore m_1-m_2=\sqrt{(m_1+m_2{)}^2-4m_1{m}_2}$
$=\sqrt{144-4}$
$=\sqrt{140}$