Michaelis and Menten's equation is an hyperbola

It is mentioned in the lecture that Michaelis and Menten's equation is an hyperbola, but there isn't a proof, here it is:

Let $ x = [s] $ and $ y = v_{0} $ as we see in the graph. Let $ m = v_{max} $ and $ k = k_{m} $ to simplify notation. We have

$ \begin{eqnarray*} y &=& \frac{mx}{k + x} \\ (k+x)y &=& mx \\ ky + xy &=& mx \\ \end{eqnarray*} $

The $ xy $ term is annoying, so we let $ x = X + Y $ and $ y = X - Y $, this is a linear transformation that simply rotate and scales, so we have

$ \begin{eqnarray*} k(X - Y) + (X + Y)(X - Y) &=& m(X + Y) \\ kX - kY + X^2 - Y^2 &=& mX + mY \\ \end{eqnarray*} $

Now we can see why it is a hyperbola - simply complete the square will lead us to the standard hyperbola.