Focal-distance identity for a hyperbola ($|d_{-}-d_{+}|=2a$)

For $x^2/a^2-y^2/b^2=1$, let $c=\sqrt{a^2+b^2}$ and place the foci of a conic at $(\pm c,0)$. On the right branch write $x=a\cosh t$, $y=b\sinh t$. Direct expansion givesSince $c>a>0$, both square roots are positive and $d_{-}-d_{+}=2a$. On the reflected left branch the signed difference reverses, while its absolute value stays $2a$. This proves the constant-difference characterization of a hyperbola and keeps track of which branch a hyperbolic function parametrization covers.