Question

The function $f(x),$ defined for $x > 0,$ is positive, differentiable, and decreasing.  The function $f(x)$ has the following property: Let $P$ be a point on the graph, and draw the tangent to the graph at $P.$  Let the tangent meet the $x$-axis and $y$-axis at $A$ and $B,$ respectively.  Then the length of chord $AB$ is constant.

Find all functions $f(x)$ that have this property.