Question

If $x^2-xy+\lambda y^2+2x-5y+2=0$ represents a pair of stright lines, find the value of $\lambda$.

Answer 1

As we know that equation of pair of straight lines is:

$a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0$

if, $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

We have the equation of straight line :

$x^2-xy+\lambda y^2+2x-5y+2=0$

Here, $a=1,b=\lambda ,h=-\frac{1}{2},g=1,f=-\frac{5}{2},c=2$.

The given equation represents a pair of straight lines.

$\therefore abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

$\Rightarrow 1\times \lambda \times 2+2\times (-\frac{5}{2})\times 1\times (-\frac{1}{2})-1\times {(-\frac{5}{2})}^2-\lambda \times 1^2-2\times {(-\frac{1}{2})}^2=0$

$\Rightarrow 2\lambda +\frac{5}{2}-\frac{25}{4}-\lambda -\frac{1}{2}=0$

$\Rightarrow \lambda =\frac{25}{4}+\frac{1}{2}-\frac{5}{2}$

$\Rightarrow \lambda =\frac{25+2-10}{4}$

$\Rightarrow \lambda =\frac{17}{4}$