Question

If the slope of one of the lines given by $a{x}^2+2hxy+b{y}^2=0$ is two times the other then

• A. $8h^2=9ab$
• B. $8h^2=9a{b}^2$
• C. $8h=9ab$
• D. $8h=9a{b}^2$

Answer 1

The correct option is A $8h^2=9ab$

For, $b\ne 0$, the given equation can be written as:-

$y^2+\frac{2h}{b}xy+\frac{a}{b}{x}^2=0$

Let the equations of lines be $y+m_{1}x=0$ and $y+m_{2}x=0$

Hence, equation of system of lines becomes:-

$(y+m_{1}x)(y+m_{2}x)=0$

$⟹y^2+(m_1+m_2)xy+m_1{m}_2{x}^2=0$

Comparing with the given equation, we get:-

$m_1+m_2=\frac{2h}{b}$ and $m_1{m}_2=\frac{a}{b}$

Now, $m_1=2m_2$

$⟹2m_2+m_2=\frac{2h}{b}$

$⟹m_2=\frac{2h}{3b}$ and hence $m_1=\frac{4h}{3b}$

Now, $m_1{m}_2=\frac{a}{b}$

$⟹\frac{8h^2}{9b^2}=\frac{a}{b}$

$⟹8h^2=9ab$

Hence, answer is option-(A).