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Assume given a point *F* in the plane, a line *d* not going through
*F*, and a positive real number *e*. The set of points *P* such that
the distance *PF* is *e* times the distance from *P* to *d* (measured
along a perpendicular) is a conic. We call *F* the **focus**, *d*
the **directrix** and *e* the eccentricity of the conic. If *e*<1 we
have an ellipse, if *e*=1 a parabola, and if *e*>1 a hyperbola
(Figure 1). This construction gives all conics
except the circle, which is a particular case of the ellipse according
to the earlier definition (we can recover it by taking the limit
*e*0).

**Figure 1:** Left: Definition of conics by means of the ratio
(eccentricity) between the distance to a point and the distance to a
line. On the left, *e*=.7; on the middle, *e*=1; on the right, *e*=2.

For any conic, a line perpendicular to *d* and passing through *F* is
an axis of symmetry. The ellipse and the hyperbola have an additional
axis of symmetry, perpendicular to the first, so there is an alternate
focus and directrix *F'* and *d'* obtained as the reflection of *F*
and *d* with respect to this axis. (By contrast, the focus and
directrix are uniquely defined for a parabola.)

The simplest analytic form for the ellipse and hyperbola is obtained when the two symmetry axes coincide with the coordinate axes. The ellipse in Figure 2 has equation

with *b*<*a*. The *x*-axis is the **major axis**, and the *y*-axis is
the **minor axis**. These names are also applied to the segments
determined on the axes by the ellipse, and to the lengths of these
segments: 2*a* for the major axis and 2*b* for the minor. The
**vertices** are the intersections of the major axis with the
ellipse, and have coordinates (*a*,0) and (-*a*,0). The distance from
the center to either **focus** is , and the sum of
the distances from a point in the ellipse to the foci is 2*a*. The
**latera recta** (in the singular, **latus rectum**) are the chords
perpendicular to the major axis and going through the foci; their
length is 2*b*/*a*. The **eccentricity** is /*a*. All
ellipses of the same eccentricity are similar; in other words, the
shape of an ellipse depends only on the ratio *b*/*a*. The distance
from the center to either **directrix** is *a*/.

**Figure 2:** Left: Ellipse with major semiaxis *a* and minor semiaxis *b*.
Here *b*/*a*=0.6.

The hyperbola in Figure 3 has equation

The *x*-axis is the **transverse axis**, and the *y*-axis is the
**conjugate axis**. The **vertices** are the intersections of the
transverse axis with the ellipse, and have coordinates (*a*,0) and
(-*a*,0). The segment thus determined, and its length 2*a*, is also
called the transverse axis, while the length 2*b* is also called the
conjugate axis. The distance from the center to either **focus** is
, and the difference between the distances from a
point in the hyperbola to the foci is 2*a*. The **latera recta** are
the chords perpendicular to the transverse axis and going through the
foci; their length is 2*b*/*a*. The **eccentricity** is
/*a*. The distance from the center to either
**directrix** is *a*/. The legs of the hyperbola
approach the **asymptotes**, lines of slope ±*b*/*a* that cross at
the center.

**Figure 3:** Left: Hyperbola with transverse semiaxis *a* and conjugate
semiaxis *b*.
Here *b*/*a*=0.4.

All hyperbolas of the same eccentricity are similar; in other words,
the shape of a hyperbola depends only on the ratio *b*/*a*. Unlike the
case of the ellipse (where the major axis, containing the foci, is
always longer than the minor axis), the two axes of a hyperbola can
have arbitrary lengths. When they have the same length, so that
*a=b*, the asymptotes are perpendicular, and *e*=, the hyperbola
is called **rectangular**.

The simplest analytic form for the parabola is obtained when the
symmetry axis coincides with one coordinate axis and the
**vertex** (the intersection of the axis with the curve) is at
the origin. The parabola
on the right has equation

where *a* is the distance from the vertex to the focus, or, which is
the same, from the vertex to the directrix. The **latus rectum** is
the chord perpendicular to the axis and going through the focus; its
length is 4*a*. All parabolas are similar: they can be made to look
identical by scaling, translation and rotation.

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.