= Hyperbola
{title2=$x^2/a^2-y^2/b^2=1$}
{wiki}
A hyperbola is a <conic section> with two unbounded branches. In suitable coordinates its equation is $x^2/a^2-y^2/b^2=1$ with $a,b>0$. The unit hyperbola is parametrized by $x=\cosh t$, $y=\sinh t$ on its right branch, relating it to the <hyperbolic functions>.
For example, $x^2/9-y^2/16=1$ has <foci of a conic> at $(\pm5,0)$, because $5^2=3^2+4^2$. Its focal-distance difference has absolute value $6$, while its <eccentricity of a conic> is $5/3>1$. At the point $(3,0)$, the focal distances are $8$ and $2$, verifying that difference. The <asymptotes> are $y=\pm4x/3$; a distant point approaches those directions while remaining on one of the two branches. The focal-distance difference, rather than a constant sum, distinguishes this <hyperbola> from an <ellipse>.
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