Thank you for reporting, we will resolve it shortly
Q.
The slopes of the lines represented by $x^2 + 2hxy + 2y^2 = 0$ are in the ratio $1 : 2$, then h equals
Straight Lines
Solution:
We know that if $m_1$ and $m_2$ are the slopes of the lines represented by $ax^2 + 2hxy + by^2 = 0,$
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 $ax^{2} + 2hxy + by^{2} = 0$,
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 $h=\pm\frac{3}{2}.$