The equations for hyperbolas are very similar to those for ellipses,
only with a minus sign.
The standard rectangular formulas for hyperbolas are $$\frac{x^2}{a^2} 
\frac{y^2}{b^2} = 1$$and$$\frac{y^2}{a^2}  \frac{x^2}{b^2}=1.$$The
first has foci at $(\pm c, 0)$, where $$c^2 =a^2 + b^2$$ (instead of
$a^2b^2$ for an ellipse), and the second has foci at $(0, \pm c)$. As
before, the ratio $c/a$ is called the eccentricity, only now it is
bigger than 1.

The hyperbola $x^2y^2=1$ opens to the sides.

The hyperbola $y^2x^2=1$ opens up and down.

Hyperbolas can also be described by a distance formula.
Amazing fact for hyperbolas: The hyperbola is the set of all points where
$L_1  L_2 = 2a$.

(This Amazing Fact can be derived almost exactly the same way
that we derived the Amazing Fact for ellipses.)
A hyperbola consists of two pieces, one with $L_1L_2 = 2a$ and one
with $L_1  L_2 = 2a$. The four ends approach the lines $y = \pm
\frac{b}{a}x$ (for the first kind of hyperbola) or $y = \pm
\frac{a}{b}x$ (for the second kind). These lines are called
asymptotes.
Example: The asymptotes of the hyperbola $x^2  y^2 = 2$ are
the diagonal lines $y = \pm x$. If you rotate this hyperbola
counterclockwise by 45 degrees, you get the curve $y = \frac{1}{x}$,
whose asymptotes are the $x$ and $y$ axes.
