Question

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

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

Answer 1

Answer: (B)

We know that if $m_1$ and $m_2$ are the slopes of the lines represented by $a{x}^2+2hxy+b{y}^2=0,$
then sum of slopes $=m_1+m_2=-\frac{2h}{b}$ and
product of slopes $=m_1{m}_2=\frac{a}{b}$.
Consider the given equation which is $x^2+2hxy+2y^2=0$
On comparing this equation with $a{x}^2+2hxy+b{y}^2=0$,
we have $a=1,2h=2h$ and $b=2$
Let the slopes be $m_1$ and $m_2$.
Given : $m_1:m_2=1:2$
Let $m_1=x$ and $m_2=2x$
$\therefore m_1+m_2=-\frac{2h}{2}\Rightarrow x+2x=-h\Rightarrow h=-3x...\left(i\right)$
and $m_1{m}_2=\frac{a}{b}\Rightarrow x.2x=\frac{1}{2}\Rightarrow x=\pm \frac{1}{2}...\left(ii\right)$
$\therefore$ From eqs. $\left(i\right)$ and $\left(ii\right)$, we have $h=\pm \frac{3}{2}.$