Question

The value of $\lambda$ if $\lambda x^2-10xy+12y^2+5x-16y-3=0$ represents a pair of straight lines is

• A. $1$
• B. $-1$
• C. $2$
• D. $-2$

Answer 1

The correct option is C

$2$

Explanation for the correct option:

Given, $\lambda x^2-10xy+12y^2+5x-16y-3=0$

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:

$$\begin{gathered} \lambda \left(12\right)(-3)+2(-8)(-5)\left(\frac{5}{2}\right)-\lambda {(-8)}^2-12{\left(\frac{5}{2}\right)}^2+3{(-5)}^2=0 \\ \Rightarrow -36\lambda +200-64\lambda -75+75=0 \\ \Rightarrow -100\lambda +200=0 \\ \Rightarrow \lambda =2 \end{gathered}$$

Therefore, the value of $\lambda$, for which the equation $\lambda x^2-10xy+12y^2+5x-16y-3=0$ represents a pair of straight lines, is $2$.

Hence, option C is the correct answer.