area of parallelogram formed by the lines y = mx , y = mx + 1, y = nx + 1 and y = nx equals?

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$$\begin{gathered} \mathrm{From} \mathrm{figure}, \\ \mathrm{A}\left(0,1\right) , \mathrm{C}\left(0,0\right) \\ \mathrm{B}: \mathrm{y}=\mathrm{mx} \mathrm{and} \mathrm{y}=\mathrm{nx}+1 \\ \mathrm{mx}=\mathrm{nx}+1 \\ \mathrm{x}=\frac{1}{\mathrm{m}-\mathrm{n}} \\ \mathrm{y}=\mathrm{mx}=\frac{\mathrm{m}}{\mathrm{m}-\mathrm{n}} \\ \mathrm{B}\left(\frac{1}{\mathrm{m}-\mathrm{n}},\frac{\mathrm{m}}{\mathrm{m}-\mathrm{n}}\right) \\ \mathrm{D}: \mathrm{y}=\mathrm{nx} \mathrm{and} \mathrm{y}=\mathrm{mx}+1 \\ \mathrm{nx}=\mathrm{mx}+1 \\ \mathrm{x}=\frac{1}{\mathrm{n}-\mathrm{m}} \\ \mathrm{y}=\mathrm{nx} \\ \mathrm{y}=\frac{\mathrm{n}}{\mathrm{n}-\mathrm{m}} \\ \mathrm{D}\left(\frac{1}{\mathrm{n}-\mathrm{m}},\frac{\mathrm{n}}{\mathrm{n}-\mathrm{m}}\right) \\ \mathrm{area} \mathrm{of} \mathrm{paralleogram} \mathrm{ABCD}=2\times \mathrm{area} \mathrm{of} △\mathrm{ABC} \\ △\mathrm{ABC} \mathrm{area}=\frac{1}{2}\left|\begin{array}{ccc}0& 1& 1\\ \frac{1}{\mathrm{m}-\mathrm{n}}& \frac{\mathrm{m}}{\mathrm{m}-\mathrm{n}}& 1\\ 0& 0& 1\end{array}\right|=\frac{1}{2}\times \left(0+0+1.\left(\frac{1}{\mathrm{m}-\mathrm{n}}-0\right)\right)=\frac{1}{2\left(\left|\mathrm{m}-\mathrm{n}\right|\right)} \left(\mathrm{using} \mathrm{determinant} \mathrm{method}\right) \\ \mathrm{hence}, \\ \mathbf{area}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{paralleogram}\mathbf{ }\mathbf{ABCD}\mathbf{=}\mathbf{2}\mathbf{\times }\mathbf{area}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{△}\mathbf{ABC}\mathbf{=}\mathbf{2}\mathbf{\times }\frac{\mathbf{1}}{\mathbf{2}\left(\mathbf{m}\mathbf{-}\mathbf{n}\right)}\mathbf{=}\frac{\mathbf{1}}{\left|\mathbf{m}\mathbf{-}\mathbf{n}\right|} \end{gathered}$$


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