Question

Find the value of $h,$ if the slopes of the lines represented by $6x^2+2hxy+y^2=0$ are in the ratio $1:2.$

Answer 1

Equation for pair of straight lines passing through origin.

$\left(y-m_{1}x\right)\times \left(y-m_{2}x\right)=0$ (where $m_1.m_2$ are slopes of two lines is $6x^2+2hxy+y^2=0$ )

$\frac{m_1}{m_2}=\frac{1}{2}$.......(1)

As per equation of pair of straight lines passing through origin

$m_1\times m_2=6$.......(2)

$\therefore (m_1+m_2)=2h$.....(3)

Now from (1) using value of $m_2$ in (2) and (3) we get,

$2m_1^2=6$

or, $m_1^2=3$......(4).

And,

$3m_1=2h$

or, $m_1=\frac{2h}{3}$.

Using the value of this $m_1$ in (4) we get,

$\frac{4h^2}{9}=3$

or, $h=\pm \frac{\sqrt{27}}{2}$.