6. In a Cartesian coordinate system $O x y$, consider the equilateral hyperbola with equation $x y = k$, with $k$ a non-zero real parameter. Let $t$ be the tangent line to the hyperbola at a point $P$ of it. Let $A$ and $B$ be the points where $t$ intersects the axes of the reference frame. Prove that the triangles $A P O$ and $B P O$ are equivalent and that their area does not depend on the choice of $P$.