The area (in sq. units) of the quadrilateral formed by two pairs of lines $l^2{x}^2-m^2{y}^2-n(lx+my)=0$  and  $l^2{x}^2-m^2{y}^2-n(lx-my)=0$  is

Given equation can be reduced in product of linear equation  $(\mathcal{l}x+my)(\mathcal{l}x-my-n)=0$
And  $(\mathcal{l}x-my)(\mathcal{l}x+my+n)=0$
$\Rightarrow \mathcal{l}x+my=0,\mathcal{l}x-my-n=0$ 
And  $\mathcal{l}x-my=0,\mathcal{l}x+my+n=0$
Area  $=\left|\frac{(c_1-d_1)(c_2-d_2)}{(a_1{b}_2-a_2{b}_1)}\right|=\left|\frac{(0+n)(0-n)}{(-\mathcal{l}m-\mathcal{l}m)}\right|=\frac{n^2}{2\left|\mathcal{l}m\right|}$