Hyperbola Research Paper Example | Topics and Well Written Essays - 750 words

Hyperbola - Research Paper Example 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 of 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 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 with each other and two parts are positioned on equal distances with each other. As the value of the eccentricity of hyperbola come closer to 1 t he 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 widens and the two ends of cup go more far with each other. 1. Mathematical And Geometric Features In Hyperbola Hyperbola has many geometric and mathematical features as that of ellipse. ... There is another axis at the centre of hyperbola which is perpendicular to that of traverse axis and is called as conjugate axis. The conjugate axis is just like a minor axis as in the case of 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 origin as well as it can be replaced to some point like (h, k), in such case of replacement of centre point the equations and calculations are made accordingly and become bit difficult to solve. The traverse axis is normally parallel to X axis (horizontal) or exactly placed on it, but in other cases the traverse axis can be shifted to Y axis (vertical) for this reason a new hyperbolic curvature is obtained. When the transverse axis is placed on X axis the conjugate axis will be placed on Y axis. Bu t when traverse axis is placed on Y axis the conjugate axis will be shifted to X axis. When traverse axis is placed on or parallel to X axis (horizontal), the ‘x’ component in standard equation of hyperbola is taken as positive. If the transverse axis is placed on or parallel to Y axis (vertical), the ‘y’ component in standard equation of hyperbola is taken as positive while the other ‘x’ component is taken as negative. 2. Hyperbolic Classification There are different types of hyperbola categorized on the basis of their orientation and characteristics. Hyperbolae centre can be placed on origin and as well as any where except origin. Hyperbole transverse axis can be parallel to X axis (horizontal axis), can be made parallel