TUYMAADA 2017 S8

Two points $A$ and $B$ are given in the plane. A point $X$ is called their preposterous midpoint if there is a Cartesian coordinate system in the plane such that the coordinates of $A$ and $B$ in this system are non-negative, the abscissa of $X$ is the geometric mean of the abscissae of $A$ and $B$ , and the ordinate of $X$ is the geometric mean of the ordinates of $A$ and $B$ . Find the locus of all the preposterous midpoints of $A$ and $B$ .

(K. Tyschu)