Analyzing The Hyperbola $\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1$ Features And Graph

Jul 10, 2025 by ADMIN 87 views

$Understanding the Hyperbola: A Detailed Exploration of ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$$

Introduction
Identifying the Equation: Hyperbola
Key Features of the Hyperbola
- 3.1. Center
- 3.2. Vertices
- 3.3. Foci
Step-by-Step Analysis
- 4.1. Determining the Center
- 4.2. Locating the Vertices
- 4.3. Calculating the Foci
Graphical Representation
Conclusion

1. Introduction

In the realm of analytic geometry, hyperbolas stand as fascinating curves with unique properties and applications. Characterized by their distinctive two-branch structure, hyperbolas are defined by a specific equation form that dictates their shape, orientation, and position in the Cartesian plane. This article delves into the detailed analysis of a particular hyperbola equation, ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ , aiming to comprehensively identify and explain its key features. We will explore the significance of each parameter in the equation, guiding you through the process of determining the center, vertices, and foci of the hyperbola. Through this exploration, we aim to provide a clear understanding of how these features define the hyperbola's graphical representation and its mathematical essence. Whether you are a student seeking to enhance your knowledge of conic sections or a mathematics enthusiast intrigued by the beauty of geometric forms, this article will serve as a valuable resource in unraveling the intricacies of hyperbolas. We will also discuss the practical applications of hyperbolas in various fields, highlighting their importance beyond theoretical mathematics.

2. Identifying the Equation: Hyperbola

The given equation ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ is a standard form equation representing a hyperbola. It is crucial to recognize this form because the structure of the equation directly reveals essential characteristics of the hyperbola. The general equation for a hyperbola centered at the origin can take two primary forms:

${\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1}$ : This form represents a hyperbola that opens horizontally, with the transverse axis along the x-axis.
${\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1}$ : This form represents a hyperbola that opens vertically, with the transverse axis along the y-axis.

In our case, the equation ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ matches the second form, indicating that this hyperbola opens vertically. The values in the denominators, ${96^2}$ and ${40^2}$ , provide critical information about the hyperbola's dimensions. Specifically, the values 96 and 40 are directly related to the distances from the center to the vertices and the shape of the hyperbola, respectively. Understanding this basic identification is the first step in dissecting the hyperbola's properties. Furthermore, the subtraction sign between the two terms is a key indicator of a hyperbola, distinguishing it from an ellipse, which has an addition sign. By recognizing the standard form, we can readily apply the formulas and concepts associated with hyperbolas to determine its specific features.

3. Key Features of the Hyperbola

To fully describe the hyperbola represented by the equation ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ , we need to identify three key features: the center, the vertices, and the foci. Each of these elements plays a crucial role in defining the hyperbola's shape and position in the coordinate plane.

3.1. Center

The center of a hyperbola is the midpoint of the line segment connecting the vertices and also the midpoint of the line segment connecting the foci. For hyperbolas in the standard form ${\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1}$ or ${\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1}$ , the center is located at the point ${(h, k)}$ . In our equation, ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ , which can be rewritten as ${\frac{(y-0)^2}{96^2} - \frac{(x-0)^2}{40^2} = 1}$ , the center is clearly at the origin, ${(0, 0)}$ . The center serves as the reference point from which all other features are measured, and its location significantly impacts the hyperbola's position on the graph. Understanding the center is essential for accurately plotting the hyperbola and interpreting its characteristics. The center also helps in determining the asymptotes of the hyperbola, which are lines that the hyperbola approaches as it extends to infinity. These asymptotes intersect at the center, providing further insight into the hyperbola's overall structure. Therefore, identifying the center is the foundational step in analyzing a hyperbola.

3.2. Vertices

The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis. The transverse axis is the axis that passes through the center and the foci. For a hyperbola in the form ${\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1}$ , the vertices are located at ((0, \pm a)). In our equation, ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ , the value of ${a}$ is 96. Therefore, the vertices are at ${(0, 96)}$ and ${(0, -96)}$ . These points are crucial because they define the endpoints of the transverse axis and determine the hyperbola's opening direction. Since the y² term is positive, the hyperbola opens vertically, with the vertices lying along the y-axis. The distance between the vertices is ${2a}$ , which is known as the length of the transverse axis. The vertices not only mark the closest points of the hyperbola to the center but also serve as key points for sketching the hyperbola's graph. Understanding the location of the vertices is essential for visualizing the hyperbola's shape and orientation in the coordinate plane. The vertices, along with the center, provide the basic framework for understanding the hyperbola's structure.

3.3. Foci

The foci of a hyperbola are two fixed points inside the two branches of the hyperbola. The distance from any point on the hyperbola to the two foci has a constant difference. The foci are located along the transverse axis, and their position is determined by the distance ${c}$ from the center, where ${c}$ is related to ${a}$ and ${b}$ by the equation ${c^2 = a^2 + b^2}$ . For the given hyperbola ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ , we have ${a = 96}$ and ${b = 40}$ . To find ${c}$ , we use the formula:

${ c^2 = a^2 + b^2 = 96^2 + 40^2 = 9216 + 1600 = 10816 }$ ${ c = \sqrt{10816} = 104 }$

Since the hyperbola opens vertically, the foci are located at ((0, \pm c)), which means they are at ${(0, 104)}$ and ${(0, -104)}$ . The foci are critical points that influence the hyperbola's shape. The distance between the foci, ${2c}$ , is a measure of the hyperbola's spread. The larger the value of ${c}$ , the wider the hyperbola. The foci are also used in the definition of a hyperbola: for any point on the hyperbola, the absolute difference in distances to the two foci is constant and equal to ${2a}$ . Understanding the foci is crucial for a complete understanding of the hyperbola's geometric properties and its applications in physics and engineering. For example, the reflective property of hyperbolas, where rays directed towards one focus are reflected towards the other focus, has practical applications in telescope design and satellite antennas.

4. Step-by-Step Analysis

Let's conduct a step-by-step analysis to identify the features of the hyperbola described by the equation ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ . This process will solidify our understanding and demonstrate how to systematically approach such problems.

4.1. Determining the Center

The first step in analyzing any hyperbola equation is to determine the center. As discussed earlier, the standard form of a hyperbola centered at ${(h, k)}$ is either ${\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1}$ or ${\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1}$ . Our equation, ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ , can be rewritten as ${\frac{(y-0)^2}{96^2} - \frac{(x-0)^2}{40^2} = 1}$ . By comparing this to the standard form, we can clearly see that ${h = 0}$ and ${k = 0}$ . Therefore, the center of the hyperbola is at ${(0, 0)}$ . This means the hyperbola is centered at the origin of the coordinate plane, simplifying subsequent calculations and graphical representation. The center serves as the reference point for all other features, such as vertices and foci, making its identification the essential first step. Additionally, the center's location provides immediate insight into the symmetry of the hyperbola, which is symmetric about both the x-axis and the y-axis in this case. Knowing the center also aids in sketching the asymptotes, which intersect at this point, further defining the hyperbola's overall shape and behavior.

4.2. Locating the Vertices

Next, we need to locate the vertices of the hyperbola. Since the equation is in the form ${\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1}$ , we know that the hyperbola opens vertically, and the vertices lie along the y-axis. The vertices are located at ((0, \pm a)), where ${a}$ is the square root of the denominator of the ${y^2}$ term. In our equation, ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ , the value of ${a^2}$ is ${96^2}$ , so ${a = 96}$ . Therefore, the vertices are at ${(0, 96)}$ and ${(0, -96)}$ . These vertices are the points where the hyperbola intersects its transverse axis, which in this case is the y-axis. The distance between the vertices, ${2a}$ or 192 units, defines the length of the transverse axis. The vertices are crucial for sketching the hyperbola's branches, as they mark the closest points of the hyperbola to the center. Understanding the location of the vertices provides a fundamental understanding of the hyperbola's orientation and shape. Moreover, the vertices are essential for determining the asymptotes, which pass through the center and guide the hyperbola's asymptotic behavior. Thus, identifying the vertices is a critical step in fully analyzing the hyperbola.

4.3. Calculating the Foci

Finally, we calculate the foci of the hyperbola. As previously discussed, the foci are located along the transverse axis, and their distance from the center is denoted by ${c}$ . The relationship between ${a}$ , ${b}$ , and ${c}$ is given by the equation ${c^2 = a^2 + b^2}$ . For our hyperbola, ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ , we have ${a = 96}$ and ${b = 40}$ . Plugging these values into the equation, we get:

${ c^2 = 96^2 + 40^2 = 9216 + 1600 = 10816 }$

Taking the square root of both sides, we find:

${ c = \sqrt{10816} = 104 }$

Since the hyperbola opens vertically, the foci are located at ((0, \pm c)), which means they are at ${(0, 104)}$ and ${(0, -104)}$ . The foci are key points that define the hyperbola's curvature and are used in the formal definition of a hyperbola: for any point on the hyperbola, the absolute difference of its distances to the two foci is constant and equal to ${2a}$ . The foci provide deeper insight into the hyperbola's geometric properties, influencing its shape and reflective characteristics. For instance, rays directed towards one focus of a hyperbolic mirror are reflected towards the other focus, a principle used in various optical instruments. The distance between the foci, ${2c}$ , indicates the spread of the hyperbola; a larger value of ${c}$ results in a wider hyperbola. Therefore, calculating the foci is a crucial step in fully understanding the hyperbola's characteristics and its applications.

5. Graphical Representation

The graphical representation of the hyperbola ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ visually confirms our analysis and provides a comprehensive understanding of its features. The hyperbola is centered at the origin ${(0, 0)}$ , which serves as the central point around which the two branches open vertically. The vertices, located at ${(0, 96)}$ and ${(0, -96)}$ , define the endpoints of the transverse axis and indicate the closest points of the hyperbola to the center. These vertices anchor the hyperbola's shape and orientation, emphasizing its vertical opening. The foci, positioned at ${(0, 104)}$ and ${(0, -104)}$ , lie further from the center than the vertices, underscoring their role in determining the hyperbola's curvature. The greater the distance of the foci from the center, the wider the hyperbola's branches. In addition to these key points, the graph also implicitly shows the asymptotes, which are lines that the hyperbola approaches as it extends to infinity. These asymptotes intersect at the center and provide a framework for sketching the hyperbola's branches accurately. The visual representation vividly illustrates the symmetry of the hyperbola about both the x-axis and the y-axis, a direct consequence of its equation form. Graphing the hyperbola not only solidifies our understanding of its geometric properties but also showcases the elegance and symmetry inherent in conic sections. Through the graph, we can readily appreciate how the center, vertices, and foci collectively define the hyperbola's shape and position in the coordinate plane.

6. Conclusion

In conclusion, the detailed analysis of the hyperbola described by the equation ${\frac{y^2}{96^2} - \frac{x^2}{40^2} = 1}$ has provided a comprehensive understanding of its key features and graphical representation. We systematically identified the center at ${(0, 0)}$ , the vertices at ${(0, 96)}$ and ${(0, -96)}$ , and the foci at ${(0, 104)}$ and ${(0, -104)}$ . Each of these elements plays a crucial role in defining the hyperbola's shape, orientation, and position in the coordinate plane. The center serves as the reference point, the vertices mark the endpoints of the transverse axis, and the foci influence the curvature of the hyperbola's branches. The graphical representation visually confirmed these features, showcasing the hyperbola's symmetry and its asymptotic behavior. This exploration underscores the importance of recognizing the standard form of a hyperbola equation and understanding how to extract meaningful information from it. The ability to identify and calculate the center, vertices, and foci is essential for both theoretical understanding and practical applications of hyperbolas. Whether in the context of mathematical studies or real-world applications such as optics and engineering, a solid grasp of these concepts is invaluable. This article has aimed to provide a clear and detailed guide to analyzing hyperbolas, equipping readers with the knowledge and skills to tackle similar problems with confidence. By breaking down the analysis into manageable steps and providing clear explanations, we hope to have demystified the intricacies of hyperbolas and highlighted their mathematical elegance.