Question

The value of $\lambda$, for which the equation $x^2-y^2-x-\lambda y-2=0$ represents a pair of straight lines, are

• A. $3,-3$
• B. $-3,1$
• C. $3,1$
• D. $-1,1$

Answer 1

The correct option is A

$3,-3$

Explanation for the correct option:

Find the value of $\lambda$.

Given, $x^2-y^2-x-\lambda y-2=0$

We know that if an equation $a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0$ is a pair of straight lines, then $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$.

So, from the given equation:

$$\begin{gathered} 1(-1)(-2)+\frac{\lambda }{2}\left(0\right)-{\left(-\frac{\lambda }{2}\right)}^2+{\left(-\frac{1}{2}\right)}^2+2\left(0\right)=0 \\ \Rightarrow 2-\frac{{\lambda }^2}{4}+\frac{1}{4}=0 \\ \Rightarrow 8-{\lambda }^2+1=0 \\ \Rightarrow {\lambda }^2-9=0 \\ \Rightarrow {\lambda }^2=9 \\ \Rightarrow \lambda =\pm 3 \end{gathered}$$

Therefore, the value of $\lambda$, for which the equation $x^2-y^2-x-\lambda y-2=0$ represents a pair of straight lines, are $\left\{3,-3\right\}$.

Hence, option A is the correct answer.