The area of the triangle formed by the lines $x^2-3xy+y^2=0andx+y+1=0$

Given Equation $x^2-3xy+y^2=0$

comparing with $a{x}^2+2hxy+b{y}^2=0$

$a=1, 2h=-3, b=1$

and $x+y+1=0$ comparing with $lx+my+n=0$

$l=1, m=1, n=1$

Area of ${∆}^{le}$

$\begin{array}{l}=\frac{n^2\sqrt{h^2-ab}}{\left|a{m}^2-2hlm+b{l}^2\right|}\\ \Rightarrow \frac{1\sqrt{\frac{9}{4}-1}}{\left|1{\left(1\right)}^2+3\left(1\right)\left(1\right)+1{\left(1\right)}^2\right|}\\ \frac{\sqrt{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{5}\Rightarrow \frac{\sqrt{5}}{2\times 5}=\frac{1}{2\sqrt{5}}\end{array}$