A Hyperbola, in Analytic Geometry - Research Paper Example

1. Introduction to Hyperbola

Hyperbola is a third type of conic in which the eccentricity is greater than one. The eccentricity is actually the criterion that defines the behaviour of a conic. Eccentricity is comprehended as the ratio of distances from a set of points of conic to a specific point (focus) to that of the line (directrix). Eccentricity is represented as ‘e’. Different conics have different ranges of eccentricity. Hence a type of conic is identified by the value of eccentricity and if the value of eccentricity is greater than one then the conic is named as hyperbola.

Hyperbola is logically very close to that ellipse in all its mathematical features. The basic and fundamental difference between the ellipse and a hyperbola is enumerated by the difference in the eccentricity value since the ellipse eccentric value is greater than zero but less than 1. Actually, this difference can be understood in 2D as for ellipse the sum of distances from foci and a point on that of ellipse is fixed. Whereas in hyperbola it is the difference in the distances from foci and a point on hyperbola is fixed.

The diagram of a hyperbola reveals the fact that a hyperbola is actually composed of two parts which are disjoined from each other and two parts are positioned at equal distances with each other. As the value of the eccentricity of the hyperbola come closer to 1 the edges of the cups of the hyperbola are lessened with each other coming closer on the other hand if the value of eccentricity increases the edges of the cup widen and the two ends of the cup go farther with each other.  

1. Mathematical And Geometric Features In Hyperbola

Hyperbola has many geometric and mathematical features as that of the ellipse. There are two fixed points that are controlling the locus of the hyperbola so that the eccentricity remains constant and greater than 1, these points are called foci. The line segment joining these foci is part of an axis called the transverse axis. The transverse axis actually intersects the two parts of the hyperbola at two points called vertices. There are two fixed lines called directrixes these lines along with focus keep the eccentricity constant. There is another axis at the centre of the hyperbola which is perpendicular to that of the traverse axis and is called as conjugate axis. The conjugate axis is just like a minor axis as in the case of an ellipse. Likewise, the transverse axis in a hyperbola is just like a major axis in the ellipse. The centre is a point across which all geometric features are located. This centre point is the intersection of the two axes i.e. the traverse axis and the conjugate axis. The centre point can be located at the origin as well as it can be replaced to some point like (h, k), in such case of the replacement of the centre point the equations and calculations are made accordingly and become a bit difficult to solve.  The traverse axis is normally parallel to the X axis (horizontal) or exactly placed on it, but in other cases, the traverse axis can be shifted to the Y axis (vertical) for this reason a new hyperbolic curvature is obtained. When the transverse axis is placed on the X axis the conjugate axis will be placed on the Y axis. But when the traverse axis is placed on the Y axis the conjugate axis will be shifted to the X axis. When the traverse axis is placed on or parallel to the X axis (horizontal), the ‘x’ component in the standard equation of the hyperbola is taken as positive. If the transverse axis is placed on or parallel to the Y axis (vertical), the ‘y’ component in the standard equation of the hyperbola is taken as positive while the other ‘x’ component is taken as negative.

1. Hyperbolic Classification

There are different types of hyperbolas categorized on the basis of their orientation and characteristics. Hyperbolae centre can be placed on the origin and as well as anywhere except the origin. Hyperbole transverse axis can be parallel to the X axis (horizontal axis), can be made parallel to the Y axis (vertical axis), and even rotated on the XY axis (45 degrees with horizontal X axis and Vertical Y axis). Thus on the basis of the above we have the following different types of hyperbolae:

1. Centre at origin and traverse axis on horizontal
2. Centre at origin and traverse axis on vertical

• Centre other than the origin and traverse axis parallel to horizontal

1. Centre other than the origin and traverse axis parallel to vertical
2. Rectangular Hyperbola

The first four categories of a hyperbola can be understood easily with their names, the fifth one is the type of hyperbola where the traverse axis is on a specific line x=y and the centre is at the origin.  In such cases, the denominators in the standard equation of the hyperbola are equal, while unlike the other hyperbolae the asymptotes of the rectangular hyperbola are laid on the X-axis (horizontal) and Y-axis (vertical). Rectangular hyperbola is also called equilateral hyperbolae. All five types of hyperbolae show mirror symmetry on both sides of the conjugate axis. The 3d hyperbola is called a hyperboloid and in daily life waste baskets are an example of hyperboloids.