The parametric equation of a hyperbola $\mathcal{H}$ is $$ P(t) = O \pm \cosh(t)\vec{u} + \sinh(t)\vec{v}. $$ Knowing the coordinates of $P(t)$, how to find $t$?
To solve a trigonometric equation $a \cos x + b \sin x = d$, there's a helpful formula: $$ a \cos x + b \sin x = \sqrt{a^2 + b^2} \cos\bigl(x - \textrm{arctan2}(b, a)\bigr). $$ Is there a similar formula in hyperbolic trigonometry?
From the known point and the expression for the hyperbola, you will get
$ \cosh(t) = a $
$ \sinh(t) = b $
for some numbers $a$ and $b$.
but $\cosh(t) = \frac{1}{2} ( e^{t}+ e^{-t} ) $ and $\sinh(t) = \frac{1}{2} (e^{t} - e^{-t} ) $
Thus
$ a + b = e^{t} $
from which $ t = \ln(a + b) $