A hyperbola is an open curve with two branches, the intersection
of a plane with both halves of a double cone.
In mathematics a hyperbola is a curve, specifically a smooth curve
that lies in a plane, which can be defined either by its geometric properties
or by the kinds of equations for which it is the solution set. A hyperbola has
two pieces, called connected components or
branches, which are mirror images of each other and resembling two infinite bows.
The hyperbola is one of the four kinds of conic
section, formed by the intersection of a plane and a cone.
The other conic sections are the parabola,
the ellipse, and the circle (the circle is a special case of the ellipse). Which conic section
is formed depends on the angle the plane makes with the axis of the cone,
compared with the angle a line on the surface of the cone makes with the axis
of the cone. If the angle between the plane and the axis is less than the angle
between the line on the cone and the axis, or if the plane is parallel to the
axis, then the conic is a hyperbola.
Nomenclature and features
The
asymptotes of the hyperbola (red curves) are shown as blue dashed lines and
intersect at the center of the hyperbola, C. The two focal points
are labeled F_{1} and F_{2}, and the thin black
line joining them is the transverse axis. The perpendicular thin black line
through the center is the conjugate axis. The two thick black lines parallel to
the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D_{1} and D_{2}.
The eccentricity e equals the ratio of the distances from a
point P on the hyperbola to one focus and its corresponding directrix
line (shown in green). The two vertices are located on the transverse axis at ±a relative
to the center. So the parameters are: a - distance from
center C to either vertex
b - length of a perpendicular segment from each vertex to the
asymptotes
c - distance from center C to either Focus
point, F_{1} and F_{2}, and
θ - angle formed by each asymptote with the transverse axis.
A
hyperbola is an open curve, meaning that it continues indefinitely to infinity,
rather than closing on itself as an ellipse does. A
hyperbola consists of two disconnected curves called
its arms or branches.
The points on the two branches that are closest to each other are
called their vertices, and
the line segment connecting them is called the transverse
axis or major axis,
corresponding to the major diameter of an ellipse. The midpoint of the
transverse axis is known as the hyperbola's center. The distance a from the center to each vertex is called the semi-major
axis. Outside of the transverse axis but on the same line are the two focal
points (foci) of the hyperbola. The line through these five points is one of the
two principal axes of the hyperbola, the other being the perpendicular
bisector of the transverse axis. The hyperbola has mirror symmetry about
its principal axes, and is also symmetric under a 180° turn about its center.
A conjugate axis of length 2b, corresponding to
the minor axis of an ellipse, is sometimes drawn on
the non-transverse principal axis; its endpoints ±b lie on the minor axis at
the height of the asymptotes over/under the hyperbola's vertices. Because of
the minus sign in some of the formulas below, it is also called the imaginary axis of the hyperbola.
If b = a,
the angle 2θ between the asymptotes equals 90° and the hyperbola is said
to be rectangular or equilateral.
In this special case, the rectangle joining the four points on the asymptotes
directly above and below the vertices is a square, since the lengths of its
sides 2a = 2b.
If the transverse axis of any hyperbola is aligned with the x-axis of a Cartesian
coordinate system and is centered on the origin, the equation of the hyperbola
can be written as
(Our solved example in mathguru.com uses
this concept)
http://en.wikipedia.org/wiki/Hyperbola
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