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.