 Hyperbolic parametrization of a hyperbola

For nonzero real $a,b$, the identity ${\cosh }^{2}t-{\sinh }^{2}t=1$ eliminates the parameter. The additional condition $x/a=\cosh t\ge 1$ is essential: the parametrization traces only the branch with the sign of $a$. Every point of that branch is obtained by $t=\mathrm{arsinh}(y/b)$. Its vertex is $(a,0)$ and its asymptotes are $y=\pm (b/a)x$. An algebraic equation alone would also include the opposite branch, which this real parametrization never reaches.