Given the line $\displaystyle y=-\frac{k}{m}x+k$, where k is the y axis intercept and m is the x axis intercept, and the function $\displaystyle y=f(x)$, where $\displaystyle f(x)=\frac{1}{x-1}$, find a formula for the scalar $\displaystyle a$ such that $\displaystyle af(\frac{x}{a})$ is always tangent to the line $\displaystyle y=-\frac{k}{m}x+k$ for any positive real k and m.

See the attached graph for an illustration. In plain English the question is: if the line moves, what do you have to do to "a" so that af(x/a) is still tangent to the line.

I will now send my answer to the moderators.

Moderator approved CB