Each squared equation represents a pair of parallel lines

The two pairs of lines are perpendicular, so the area is the product of the two distances between parallel lines

Note that each of the equations yields two parallel lines.

$$(x + ay)^2 = 4a^2$$ results in the two lines $$x + ay - 2a = 0$$ and $$x + ay + 2a = 0.$$ Both of these lines have a slope of $-\dfrac{1}{a}.$

Similarly, $$(ax-y)^2 = a^2$$ results in the lines $$ax - y - a = 0$$ and $$ax - y + a = 0.$$ These lines have slope $a.$

Note that each pair of lines is perpendicular to the other pair of lines. This shows that the equations form a rectangle.

Recall that the formula for the distance $d$ between two parallel lines $$\begin{cases} Ax+By+C_1=0 \\ Ax+By+C_2=0 \end{cases}$$ is $$d = \dfrac{\mid C_2 - C_1 \mid}{\sqrt{A^2 + B^2}}.$$

Using this formula, we get that the distance between the first pair of lines is $$\dfrac{4a}{\sqrt{a^2 + 1}}.$$ Similarly, the distance between the second pair of lines is $$\dfrac{2a}{\sqrt{a^2 + 1}}.$$

These are the side lengths of the rectangle. Multiplying yields the area $$\dfrac{8a^2}{a^2 + 1}.$$

Thus, D is the correct answer.