Question

If the equation $12x^2-10xy+2y^2+11x-5y+\lambda =0$ represents a pair of straight lines, then the value of $22{\lambda }^2+361\lambda +582$ is equal to

Answer 1

$a=12,h=5,g=11/2,b=2,f=5/2,c=\lambda$
Since the equation represents a pair of straight lines
$\left|\begin{array}{ccc}12& -5& 11/2\\ -5& 2& -5/2\\ 11/2& -5/2& \lambda \end{array}\right|=0$
$\Rightarrow -\lambda -\frac{25}{4}+\frac{33}{4}=0\Rightarrow \lambda =2$
$221{\lambda }^2+361\lambda +582=884+722+582=2188$