= Focal-distance identity for a hyperbola
{title2=$|d_--d_+|=2a$}
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 gives
$$
d_-^2=(x+c)^2+y^2=(c\cosh t+a)^2,\qquad
d_+^2=(x-c)^2+y^2=(c\cosh t-a)^2.
$$
Since $c>a>0$, 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.
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