A moving line intersects the lines $x + y = 0$ and $x - y = 0$ at the points $A$ and $B$ such that the area of the triangle with vertices $(0,0)$, $A$ and $B$ has a constant area $C$. The locus of the midpoint of $AB$ is given by the equation\\
(A) $\left(x^2 + y^2\right)^2 = C^2$\\
(B) $\left(x^2 - y^2\right)^2 = C^2$\\
(C) $(x + y)^2 = C^2$\\
(D) $(x - y)^2 = C^2$.