# line, function, moving tangency

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