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For $ab\ne 0$, the hyperbolic identity gives $x^2/a^2-y^2/b^2=1$. The hyperbolic parametrization of a hyperbola also imposes $x/a=\cosh \theta \ge 1$, so the exact traced locus isIt is only one branch: the right branch for $a>0$ and the left branch for $a<0$. Its vertex is $(a,0)$ and its asymptotes are $y=\pm (b/a)x$. If $a=0$, $b\ne 0$, the entire vertical axis is traced. If $b=0$, $a\ne 0$, the locus is the ray $y=0$, $x/a\ge 1$. Both zero give the origin.