Question

If $\lambda x^2+10xy+3y^2-15x-21y+18=0$ represents a pair of straight lines. Then, the value of $\lambda$ is?

• A. $-3$
• B. $3$
• C. $4$
• D. $-4$

Answer 1

The correct option is A $-3$

For the equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ to represent pair of straight lines, the condition is $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0,h^2\ge ab,g^2\ge ac,f^2\ge bc$

The given equation is $\lambda x^2+10xy+3y^2-15x-21y+18=0$

so, $a=\lambda ,h=5,b=3,g=\frac{-15}{2},f=\frac{-21}{2},c=18$

$\lambda \cdot 3.18+2\cdot \left(\frac{-21}{2}\right)\left(\frac{-15}{2}\right)\cdot 5-\lambda \left(\frac{441}{4}\right)-3\left(\frac{225}{4}\right)-18\times 25=0$

$\Rightarrow 54\lambda +787.5-11.25\lambda -168.75-18\times 25=0$

$\Rightarrow 56.25\lambda =-168.75$

$\Rightarrow \lambda =-3$