I'm trying to show algebraically that a Linear Fractional Transformation of the form $$f(x)=\frac{(ax+b)}{(cx+d)}$$ can be written as hyperbolas of the form $$(x-h)(y-k)=m$$ I started by expanding the hyperbola equation to get $$xy-hy-kx+hk=m$$ and then manipulated the LFT equation into $$cxy-ax+d=b$$ This looks promising because all I need is another y term in the LFT equation and I can divide all constants by c and I will have something in the same form as the hyperbola equation but I have no clue how to get an extra y term.
The other consideration I had was to set h=0. If I did this I would be able to get the hyperbola equation into the form $$xy-kx=m$$ and the LFT into the form $$cxy-ax=b-d$$ Is there a different way to show they are same without having to set h=0?
$$cxy-ax+d=b$$ should be $$cxy-ax+dy=b$$
$$c^2xy-acx+cdy=bc$$
$$ (cx+d)(cy-a) + ad = bc $$
$$ (cx+d)(cy-a) = bc - ad $$