An ellipse is the set of points the sum of whose distances from two fixed points (the foci) is constant. A hyperbola is the set of points the difference of whose distances from two foci is constant. One word changes — sum versus difference — and a closed oval becomes a pair of opening branches.
Ellipse (foci on the $x$-axis, $a > b$):
Here $2a$ is the major axis and $2b$ the minor axis. The foci sit at $(\pm c, 0)$ where $c = \sqrt{a^2 - b^2}$, and the vertices (endpoints of the major axis) are $(\pm a, 0)$.
Hyperbola (foci on the $x$-axis):
The vertices are $(\pm a, 0)$, the foci are $(\pm c, 0)$ with $c = \sqrt{a^2 + b^2}$, and the two branches approach the slanted asymptotes $y = \pm \dfrac{b}{a}x$.
For both curves the shape is summarised by the eccentricity $e = \dfrac{c}{a}$, and both share the same latus-rectum formula:
| Parameter | Ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ | Hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ |
|---|---|---|
| Defining condition | sum of focal distances $= 2a$ | difference of focal distances $= 2a$ |
| Vertices | $(\pm a, 0)$ | $(\pm a, 0)$ |
| Foci | $(\pm c, 0)$ | $(\pm c, 0)$ |
| Relation | $c = \sqrt{a^2 - b^2}$ | $c = \sqrt{a^2 + b^2}$ |
| Eccentricity | $0 < e < 1$ | $e > 1$ |
| Latus rectum | $\dfrac{2b^2}{a}$ | $\dfrac{2b^2}{a}$ |
The lone difference in the $c$-relations — a minus for the ellipse, a plus for the hyperbola — is exactly what pushes the eccentricity below $1$ in one case and above $1$ in the other.
Deeper Insight — eccentricity is the dial that turns one conic into another: Every conic in this chapter is a slice of a double cone, and a single number, the eccentricity $e$, records how steeply the slicing plane is tilted. A circle is the perfectly level cut with $e = 0$; tilt a little and you get an ellipse with $0 < e < 1$; tilt until the plane runs parallel to the cone's side and the closed oval breaks open into a parabola with $e = 1$; tilt further still and you cut both nappes of the cone, producing a hyperbola with $e > 1$. This is why ellipses, parabolas and hyperbolas are not three unrelated curves but three settings of the same dial — and why $e$ appears in the focus-directrix description of all of them ($PF = e \cdot PM$). The sign flip in $c = \sqrt{a^2 \mp b^2}$ is the algebraic shadow of this geometry: for an ellipse the foci must sit inside, so $c < a$ and $e < 1$; for a hyperbola the foci lie beyond the vertices, so $c > a$ and $e > 1$. Hold on to the cone picture and the entire chapter reads as one story rather than a list of formulae.