June 24, 2024

Hyperbola and its Applications

Two curves that are like infinite bows are known as hyperbolas. The word ‘hyperbola’ has been derived from a Greek word that signifies ‘overthrown or excessive’. There is a fact that you must know, an orbit of space can sometimes be a hyperbola, using a technique known as a gravitational slingshot. If this happens then the route of spacecraft is known as hyperbola. This topic seems to be a bit complicated and tough, but no worries. In this article, we will try to grasp significant concepts related to hyperbola such as how the hyperbola arises, applications, and some examples related to a hyperbola. 

How Does the Hyperbola Arise?

  1. Hyperbola arises as the curve representing the function in the cartesian plane. 
  2. It also arises as the path followed by the tip of a sundial which is also its one of the applications. 
  3. Hyperbola arises as the path of a single apparition comet. 
  4. It arises in radio navigation, when the differences between distances to two points, but not the distances themselves, can be determined.

Applications of a Hyperbola

  1. Sundials: Hyperbolas can be seen in many sundials. The sun goes on revolving around a circle on the celestial sphere, and its rays striking the point on sundial traces out a cone of light. A conic section is formed by the intersection of this cone with the horizontal plane of the ground. The collection of hyperbolas in Greek was termed pelekinon which means a double-bladed axe. 
  2. Multilateration: For solving multilateration problems, a hyperbola is used. It is the task of locating a point from the differences in its distances to the given points. Such problems are significant in navigating, particularly on ships, it can locate the distances using LORAN or GPS transmitters. 
  3. Biochemistry: In pharmacology and biochemistry, equations such as the hill equation and the Hill-Langmuir equation respectively describe biological responses. They are both regarded as a rectangular hyperbola. 
  4. Used in Making Designs and in Mathematics: The shape of a hyperbola is extensively used in the design of bridges. It also plays a significant role in calculus because of the remarkable properties of areas under the curve and also in some exponential functions. 

Formula of Hyperbola

Basically, there are two standard equations of hyperbola. These equations are based on the transverse and the conjugate axis of each of the parabola respectively. The standard equation for the transverse axis is x.x/a.a – y.y/b.b = 1, where the transverse axis lies on the x-axis and conjugates on the y-axis. The standard equation for the conjugate axis is y.y/a.a – x.x/b.b = 1. where the transverse axis lies on the y-axis and the conjugate axis on x-axis. Let us see some examples related to hyperbola in order to make this concept clearer to you. 

Some Examples Related to Hyperbola

  •  Find the position of the point (6, – 5) relative to the hyperbola x.x/9 – y.y/25 = 1.                
  • Solution: The given equation is of the hyperbola is x.x/9 – y.y/25 = 1

 According to the given problem,

   6.6/9 – (-5.-5)/25 = 1

   36/9 – 25/25 = 1

   4 – 1 – 1= 2 > 0. Therefore, the point (6, – 5) lies inside the hyperbola x.x/9 – y.y/25 =1

  • Find the lengths of transverse axis and conjugate axis of the following hyperbola 16x.x– 9y.y = −144 using the hyperbola formula. 
  • Solution: The equation 16x.x– 9y.y= −144 can be written as (x.x / 9) – (y.y / 16) = −1. 
  • This is of the form (x.x/a.a) – (y.y/ b.b) = −1

a.a= 9, b.b= 16

a=3, b=4.

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