The equation we’ll be covering today is a mathematical representation of the geometrical figure called the hyperbola. The hyperbola is a part of the family of geometrical figures called the conical sections, and it closely resembles the letter x in its graphical representation. For the study of hyperbola or any geometrical figure for that matter, a basic understanding of the concepts of geometry is a must. Having a basic understanding of these concepts will enable you to understand the terminology and concepts being discussed here quickly and in a more efficient manner. Here is a great beginner’s course about the basic concepts of geometry which will give you a good quick review about the basics and will prepare you in a better way for the concepts which are going to be discussed here.
There are two methods to write the equation of a hyperbola depending upon the orientation of the transverse axis. First of all we are going to discuss these two methods.
Horizontal Transverse Axis
x²/a² − y²/b² = 1
This this equation, the transverse axis of the hyperbola is the horizontal axis or the x-axis in the x-y plane. Here, the center would be the origin and the vertices would be (−a,0) and (a,0).
Vertical Transverse Axis
y²/a² − x²/b² = 1
Here the transverse axis of the hyperbola is the vertical axis or the y-axis in the x-y plane. Here the center would be the origin and the vertices will be (0,a) and (0,−a).
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The study of both the types of equations of the hyperbola is done essentially in the same way and here we are going to study the different aspects of the equation of the hyperbola for the equation with the horizontal transverse axis. Similar rules apply for the equation with vertical transverse axis and it can be studied using similar methods by making the appropriate changes.
Vertices of the Hyperbola
The vertices of the hyperbola are the two points (−a,0) and (a,0). These points lie on the transverse axis of the hyperbola.
Focus of the Hyperbola
The foci of the hyperbola are two points which lie on the transverse axis such that the difference of the distances of the two foci from any point on the hyperbola is a constant equal to 2a. Here, 2a is the distance between the two vertices of a hyperbola.
The line passing through the center, the two vertices and the two foci of the hyperbola is known as the transverse axis of a hyperbola. The orientation of the transverse axis depends the overall orientation of the hyperbola.
Asymptotes are the two intersecting lines which intersect at and pass through the center of the hyperbola. The most interesting characteristic of the asymptotes is that the hyperbolas keep coming closer to touching these lines but never touch them. Here is some useful information about asymptotes which will help you in understanding the concept in a better way.
The Directrix and Eccentricity
The directrix of a hyperbola is a line from which the distance of any point on the hyperbola is calculated and let this distance be d. Then the distance of that point from the corresponding focus is also taken and let it be l. Then the ratio of d and l will be a constant which will be greater than 1 and this constant is known as the eccentricity. The eccentricity determines the geometry of the hyperbola.
The General Equation of the Hyperbola
Till now we have discussed the equation of hyperbola for which the center was the origin in the Cartesian co-ordinate system. But now we are going to discuss the general for of the equation of a hyperbola.
The general form of the equation of the hyperbola is given as:
(x−h)²/a² − (y−k)²/b² = 1
Here the center of the hyperbola is given as (h,k) and the rest of the concepts remain same as above. This equation helps us in studying a hyperbola situated anywhere in the Cartesian co-ordinate system.
Here we have mentioned the algebraic equations of the hyperbola, but a similar study can also be made using the concepts of calculus. To better understand the concept of a hyperbola and to further enhance your knowledge about the related concepts of calculus, you can check out this very useful course about the concepts of geometry with an approach through calculus which will certainly take your knowledge about the concepts of the hyperbola to an altogether new level.
The concept of hyperbola finds applications in a number of areas of applied mathematics and one of the most important area of application of this concept is in calculus in areas like integral calculus. The analysis of a hyperbola becomes somewhat easier using calculus as well as this concept can also be used in a more practical sense by the use of the concepts of calculus. Here at Udemy, we have a great course which discusses the applications and usage of the concept of Hyperbola through calculus. This course discusses all the important aspects in a very informative manner. Check out this amazing course to cap-off your study of the equation of the hyperbola and its related concepts in the most efficient manner possible.