A curiosity - the reciprocal triangle

I was absent-mindedly solving simple calculus problems on Brilliant.org this afternoon when I stumbled across an interesting one about the reciprocal function. The question asked the user to prove that the area enclosed by the coordinate axes and the tangent to the curve at any point on the curve is constant, and to find this area. This is a basic proof of this unusual property:

The logical extension of this observation is to see what happens when the function is manipulated. First, imagine a reciprocal function of order n:

This sets nicely into context how unique the basic reciprocal function is - the k in the numerator will only be cancelled when n = 1.

Next, consider a function with a linear stretch factor a in the y-direction (which, due to the symmetry of the function, is the same as a stretch factor ¹/_a in the x-direction):

This is of mild interest too - the original problem took a = 1, such that the area of the triangle was 2.

Finally, consider a function translated by a units in the positive x-direction, and b units in the positive y-direction:

Clearly, as the sketch demonstrates, the translation of the curve disrupts the symmetry of the curve about the x and y axes, such that there is no longer a triangle of constant area. Trivially, it could easily be proven that there is a region of constant area, with the area under the tangent within the limits x > a and y > b, since these asymptotes effectively become the new x and y axes within the translated reference frame of the new curve. Furthermore it can be seen that the constant-area is reinstated when a = b = 0, such that k cancels out in the numerator and denominator to leave the previous area of 2.