Ellipses and hyperbolas are usually defined using two foci, but they
can also be defined using a focus and a directrix.

Definitions of conic sections: Let $L$ be the distance to the focus
and let $D$ be the distance to the directrix. Pick a constant $e>0$.
The set of all points with $L=eD$ is

An ellipse of eccentricity $e$ if $e<1$,

A parabola if $e=1$,

A hyperbola of eccentricity $e$ if $e>1$.

For instance, suppose that we have a conic section with focus at the
origin, directrix at $y=-1$, and eccentricity $e \ne 1$. Then
\begin{eqnarray*} \sqrt{x^2 + y^2} &=& e|y+1| \cr x^2 + y^2 & = & e^2
(y^2 + 2 y + 1) \cr x^2 + (1-e^2)y^2 -2e^2 y& = & e^2
. \end{eqnarray*}
Notice that the coefficient of $y^2$ in the last equation is positive
if $e<1$, giving us an ellipse, and is negative when $e>1$, giving us
a hyperbola. After completing the square and applying some more
algebraic manipulations, we can put the equation in standard form:
$$\left ( \frac{1-e^2}{e^2}\right) x^2 + \left(
\frac{(1-e^2)^2}{e^2}\right )\left ( y - \frac{e^2}{1-e^2}\right )^2 =
1.$$

If $e<1$, this is an vertically aligned ("tall and skinny") ellipse
with center at $(0,\frac{e^2}{1-e^2})$, with $a = \frac{e}{1-e^2}$,
$b=\frac{e}{\sqrt{1-e^2}}$ and $c=\frac{e^2}{1-e^2}$ and eccentricity
$e=c/a$. As $e \to 1$, the center and the size of the ellipse both go
to infinity.

If $e>1$ the situation is analogous. Our curve is then
a hyperbola of with center at $(0,-\frac{e^2}{e^2-1})$,
with $a = \frac{e}{e^2-1}$,
$b=\frac{e}{\sqrt{e^2-1}}$ and $c=\frac{e^2}{e^2-1}$ and eccentricity
$e=c/a$. As $e \to 1$, the center and the size of the hyperbola both go
to infinity.

The following video describes — without equations —
what happens to a conic section as the eccentricity is increased from 0 to 1,
and then beyond 1.