Question

The slopes of the lines represented by $x^{2}2hxy+2y^2=0$ are in the ratio $1:2$ , then it equals

• A. $\pm \frac{1}{2}$
• B. $\pm \frac{3}{2}$
• C. $\pm 1$
• D. $\pm 3$

Answer 1

Answer: (B)

If $m_1$ and $m_2$ are slopes of the lines represented by $a{x}^2+2hxy+b{y}^2=0$, then
$m_1+m_2=-\frac{2h}{b}$ and $m_1{m}_2=\frac{a}{b}$
The given equation is
$x^2+2hxy+2y^2=0$
On comparing this equation with
$a{x}^2+2hxy+b{y}^2=0$, we get
$a=1$, $2h=2h$ and $b=2$
Let the slopes of lines are $m_1$ and $m_2$
$\therefore m_1:m_2=1:2$
Let $m_1=m$ and $m_2=2m$
$\therefore m_1+m_2=-\frac{2h}{2}$
$\Rightarrow m+2m=-h$
$\Rightarrow h=-3m\dots \left(i\right)$
and $m_1{m}_2=\frac{a}{b}$
$\Rightarrow m\cdot 2m=\frac{1}{2}$
$\Rightarrow m=\pm \frac{1}{2}\dots \left(ii\right)$
From Eqs. $(i)$ and $(ii)$, we get
$h=\pm \frac{3}{2}$