Olympiad Geometry · Superchallenge problem

Two perpendicular lines pass through $P(3, 4)$. The first meets the $x$-axis at $A$ and the second meets the $y$-axis at $B$. As the pair of lines turns about $P$ (staying perpendicular), the midpoint of $\overline{AB}$ moves along a fixed line. Find the area of the triangle that this fixed line forms with the two coordinate axes.