**History**

In Renaissance, Kepler's law of planetary motion, Descarte and Fermat's coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, Pascal pushed conics to a high level. Many later mathematicians have also made contribution to conics, especially in the development of projective geometry where conics are fundamental objects as circles in Greek geometry. Among the contributors, we may find Newton, Dandelin, Gergonne, Poncelet, Brianchon, Dupin, Chasles, and Steiner. Conic sections is a rich classic topic that has spurred many developments in the history of mathematics.

Menaechmus first studied a special case of the hyperbola. This special case was xy = ab where the asymptotes are at right angles and this particular form of the hyperbola is called a rectangle hyperbola.

Euclid and Arisaeus wrote about the general hyperbola but only studied one branch of it while Apollonius who was the first to study the two branches of the hyperbola gave the hyperbola. The focus and directrix of a hyperbola were considered by Pappus.

Exploration about Mathematician: Apollonius

**Lesson I: Introduction**

Hyperbola describes a family of curves. Together
with ellipse and parabola, they make up the conic sections. Hyperbola
is two-branched open curve produced by the intersection of a circular
cone and a plane that cuts both nappes (see Figure 2.) of a cone.
of a cone. [A cone is a __pyramid__ with a circular __cross
section__ ] A degenerate hyperbola (two intersecting lines)
is formed by the intersection of a circular cone and a plane that
cuts both nappes of the cone through the apex.

**Lesson II: Definition and geometric construction
of a hyperbola**

Hyperbola is defined as the *locus* of
points P (x, y) such that the difference of the distance from
P to two fixed points F1(-c, 0) and F2(c, 0) that is called *foci*
are constant. The midpoint of the two foci points F1 and F2 is
called the *center* of a hyperbola.

That the x-intercept of locus P (x, y) is at (-a, 0) and (a, 0) gives

So the constant is , that is twice the distance between the x-intercepts.

Another definition is that as the path (*locus*)
of a point moving so that the ratio of the distance from a fixed
point F (the *focus*) to the distance from a fixed vertical
line (the *directrix*) is a constant, greater than one. (The
hyperbola, however, because of its symmetry, has two foci.)

If we let d be the distance from the center on which the directrix lies and R ratio, then we have

**Lesson III: Analytic equation of a hyperbola**

From the first definition we can draw the equation of a hyperbola (See the Figure 3):

Since

Squaring both sides, we get

Again, divide both sides of this equation by

then we obtain

Since

substitute with ,

we have

which is called the standard equation of a hyperbola.

**Lesson IV: Properties of a hyperbola**

*Eccentricity e *can
be, in verbal, explained as the fraction of the distance to the
semimajor axis at which the focus lies, where c is the distance
from the center of the conic section to the focus. Let the distance
between foci be 2c, then e (always bigger than 1) is defined as

The *eccentricity* e describes the "flatness"
of the hyperbola.

Asymtotes of a hyperbola is defined as a curve approaching a given curve arbitrarily closely.

You can get the equation of asymptotes from the standard equation of a hyperbola

In the final equations, goes to 0 if goes to infinite. According to this, the hyperbola becomes close to one of the equations,

We called these equations

as the asymptotes. As a result, the asymptotes can be written as

**Lesson V: Rectangular Hyperbola**

When the asymptotes of a hyperbola are perpendicular,
the hyperbola is called *rectangular*. This occurs when the
semimajor and semiminor axes are equal. This also means that

Further, it gives

**Lesson VI: Parametric and polar equations
of a hyperbola**

The parametric equations of a hyperbola are:

The polar equation of a hyperbola with the center at a focus is:

**Lesson
VII: In-class Worksheet**

**References**

http://mathworld.wolfram.com/Hyperbola.html

http://xahlee.org/SpecialPlaneCurves_dir/Hyperbola_dir/hyperbola.html