imc

Recall that for constants $A,B,C_1$ and $C_2,$ the distance $d$ between parallel lines is $$d=\frac{\left|C_2-C_1\right|}{\sqrt{A^2+B^2}}.$$ From this formula: * The distance between lines $(1)$ and $(2)$ is $\frac{4a}{\sqrt{1+a^2}},$ the length of this rectangle. * The distance between lines $(1\ast )$ and $(2\ast )$ is $\frac{2a}{\sqrt{a^2+1}},$ the width of this rectangle. The area we seek is $$\frac{4a}{\sqrt{1+a^2}}\cdot \frac{2a}{\sqrt{a^2+1}}=\boxed{\frac{8a^2}{a^2+1}}.$$