Chapter 4: Geometry

FIGURE 4.17
Hyperbola with transverse semiaxis and conjugate semiaxis . Here ¼.
¼
Ý
´¼µ
¼
Ç
Ü
´ ¼µ
The Ü-axis is the transverse axis, and the Ý-axis is the conjugate axis. The
vertices are the intersections of the transverse axis with the hyperbola and have coordinates
´ ¼µ and ´ ¼µ. The segment thus determined, or its length ¾, is also
called the transverse axis, while the length ¾ is also called the conjugate axis. The
distance from the center to either focus is Ô ¾ · ¾ , and the difference between the
distances from a point in the hyperbola to the foci is ¾. The latera recta are the
chords perpendicular to the transverse axis and going through the foci; their length
is ¾ ¾ . The eccentricity is Ô ¾ · ¾ . The distance from the center to either
directrix is ¾ Ô ¾ · ¾ . The legs of the hyperbola approach the asymptotes, lines
of slope ¦ that cross at the center.
All hyperbolas of the same eccentricity are similar; in other words, the shape
of a hyperbola depends only on the ratio . 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 Ô have the same length, so
that , the asymptotes are perpendicular, and ¾, the hyperbola is called
rectangular.
The simplest analytic form for the parabola is obtained
when the axis of symmetry coincides with one coordinate
axis, and the vertex (the intersection of the axis with the Ý
curve) is at the origin. The equation of the parabola on the
right is
Ý ¾ Ü (4.6.3)
´ ¼µ
where is the distance from the vertex to the focus, or, which
Ü
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 . All parabolas are similar: they can be
made identical by scaling, translation, and rotation.
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