Question

If the slope of one of the line $x^2+kxy+2y^2=0$ is twice that of the other, then the value of $k$ is

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

Answer 1

The correct option is A $\pm 3$

Given: Equation of the pair of lines

$x^2+kxy+2y^2=0\Rightarrow y^2+\frac{k}{2}xy+\frac{x^2}{2}=0$ ...(1)

and slope of one line $\left(m_1\right)=2m_2$ ..... (where $m_2$ is the slope of the other line$)$.

We know that the standard equation of the pair of lines having slopes $m_1$ and $m_2$ and passing through the origin is

$y^2-(m_1+m_2)xy+\left(m_1{m}_2\right)x^2=0.$

Comparing equation (1) with standard equation, we get

$(m_1+m_2)=-\frac{k}{2}\Rightarrow (2m_2+m_2)=-\frac{k}{2}\Rightarrow 3m_2=-\frac{k}{2}\Rightarrow m_2=-\frac{k}{6}$ ...(2)

and $(m_1\times m_2)=\frac{1}{2}\Rightarrow (2m_2\times m_2)=\frac{1}{2}\Rightarrow 2m_2^2=\frac{1}{2}\Rightarrow m_2^2=\frac{1}{4}$ ...(3)

Substituting the value of $m_2$ from Eq. $\left(2\right)$ into $\left(3\right)$, we get

${(-\frac{k}{6})}^2=\frac{1}{4}\Rightarrow k^2=9\Rightarrow k=\pm 3.$