Hyperbola Foci Calculation A Step By Step Guide

Introduction to Hyperbolas

Hyperbolas are fascinating conic sections that appear in various fields of mathematics and physics. Understanding the properties of hyperbolas, such as their foci, is crucial for solving related problems. This article delves into the specifics of identifying the foci of a hyperbola given its equation. We will dissect the equation $\frac{(x-14)^2}{162} - \frac{(y-2)^2}{302} = 1$, step-by-step, to pinpoint the correct foci. Let's embark on this mathematical journey to master hyperbola foci and their calculations.

Hyperbola Definition

A hyperbola is defined as the set of all points in a plane such that the absolute difference of the distances between two fixed points, called foci, is constant. This definition gives rise to the hyperbola's unique shape, characterized by two branches that open away from each other. The line passing through the foci intersects the hyperbola at two points called vertices, and the midpoint of the segment connecting the foci is the center of the hyperbola. The standard form equation of a hyperbola helps us identify these key features and understand the hyperbola's orientation and dimensions.

Key Components of a Hyperbola

To effectively work with hyperbolas, it's essential to understand its components. The center, vertices, foci, and asymptotes play critical roles in defining the hyperbola's shape and position. The center is the midpoint between the foci. The vertices are the points where the hyperbola intersects its major axis, which is the line passing through the foci. The foci are the two fixed points used in the definition of the hyperbola. The asymptotes are the lines that the hyperbola approaches as it extends to infinity. Understanding these components allows us to translate the algebraic equation of a hyperbola into its geometric representation, making problem-solving more intuitive and accurate.

Standard Form of Hyperbola Equations

The standard form equation of a hyperbola is a powerful tool for extracting information about its key features. There are two primary forms, depending on whether the hyperbola opens horizontally or vertically. For a hyperbola that opens horizontally, the standard form is $\frac{(x-h)^2}{a2} - \frac{(y-k)^2}{b2} = 1$, where (h, k) is the center, 'a' is the distance from the center to each vertex along the horizontal axis, and 'b' is related to the distance along the conjugate axis. For a hyperbola that opens vertically, the standard form is $\frac{(y-k)^2}{a2} - \frac{(x-h)^2}{b2} = 1$, with 'a' representing the distance from the center to each vertex along the vertical axis. The distance from the center to each focus, denoted as 'c', is given by the relationship $c^2 = a^2 + b^2$. By recognizing these forms and understanding the roles of a, b, and c, we can quickly determine the center, vertices, and foci of any hyperbola given its equation.

Analyzing the Given Equation

Identifying the Hyperbola's Orientation

The given equation is $rac{(x-14)^2}{162} - \frac{(y-2)^2}{302} = 1$. To identify the hyperbola's orientation, we examine which term is positive. In this case, the term with $(x-14)^2$ is positive, indicating that the hyperbola opens horizontally. This means that the foci and vertices lie along a horizontal line. Recognizing the orientation is the first step in determining the correct parameters for the hyperbola, such as the values of a, b, and the coordinates of the center. The horizontal orientation significantly influences how we calculate the foci and vertices, as they will be aligned along the x-axis relative to the center.

Determining the Center (h, k)

The center of the hyperbola is represented by the coordinates (h, k) in the standard form equation. By comparing the given equation $\frac{(x-14)^2}{162} - \frac{(y-2)^2}{302} = 1$ to the standard form $\frac{(x-h)^2}{a2} - \frac{(y-k)^2}{b2} = 1$, we can identify the values of h and k. Here, h = 14 and k = 2. Therefore, the center of the hyperbola is (14, 2). Pinpointing the center is crucial as it serves as the reference point for locating the vertices and foci. All these points are symmetrically placed around the center, making it a fundamental element in the hyperbola's geometry.

Finding 'a' and 'b' Values

The values 'a' and 'b' are crucial for determining the shape and dimensions of the hyperbola. In the equation $\frac{(x-14)^2}{162} - \frac{(y-2)^2}{302} = 1$, the denominators of the squared terms give us the values of $a^2$ and $b^2$. Specifically, $a^2 = 16^2$ and $b^2 = 30^2$. Taking the square root of these values, we find a = 16 and b = 30. 'a' represents the distance from the center to the vertices along the transverse axis, while 'b' is related to the distance along the conjugate axis. These values are essential for calculating the distance 'c' from the center to the foci, which is a key step in identifying the foci coordinates. Understanding 'a' and 'b' values helps in visualizing the hyperbola's stretch and shape, making it easier to solve related problems.

Calculating the Foci

Using the Relationship $c^2 = a^2 + b^2$

To find the foci of the hyperbola, we need to calculate the distance 'c' from the center to each focus. The relationship $c^2 = a^2 + b^2$ is fundamental in this calculation. We already found that a = 16 and b = 30. Substituting these values into the equation, we get $c^2 = 16^2 + 30^2 = 256 + 900 = 1156$. Taking the square root of both sides, we find c = $\sqrt{1156}$ = 34. This value 'c' represents the linear eccentricity of the hyperbola, and it is crucial for determining the location of the foci relative to the center. The formula $c^2 = a^2 + b^2$ is a cornerstone in hyperbola geometry, linking the distances from the center to the vertices and foci, and mastering this relationship is essential for solving hyperbola-related problems.

Determining the Foci Coordinates

Now that we have the value of c = 34 and the center (14, 2), we can determine the coordinates of the foci. Since the hyperbola opens horizontally, the foci lie along the horizontal axis, which means we will add and subtract 'c' from the x-coordinate of the center. The foci are located at (h ± c, k). Thus, the foci are at (14 + 34, 2) and (14 - 34, 2), which simplify to (48, 2) and (-20, 2). These coordinates represent the two fixed points that define the hyperbola, and their accurate calculation is critical for understanding the hyperbola's properties. Locating the foci coordinates is a culmination of understanding the hyperbola's orientation, center, and the relationship between a, b, and c, showcasing a comprehensive grasp of hyperbola geometry.

Final Answer

Selecting the Correct Option

Based on our calculations, the foci of the hyperbola are at (-20, 2) and (48, 2). Now, we compare these coordinates with the given options:

A. (-20, 2) and (48, 2) B. (-16, 2) and (44, 2) C. (-2, 2) and (30, 2) D. (14, -32) and (14, 36)

Clearly, option A matches our calculated foci coordinates. Therefore, option A is the correct answer. This final step reinforces the accuracy of our step-by-step calculations and demonstrates a clear understanding of how to apply hyperbola properties to solve problems. Selecting the correct option is the ultimate validation of our analytical process, confirming our ability to navigate through the complexities of hyperbola geometry.

Conclusion

In this article, we successfully identified the foci of the hyperbola represented by the equation $\frac{(x-14)^2}{162} - \frac{(y-2)^2}{302} = 1$. We began by understanding the definition and key components of a hyperbola, including its center, vertices, and foci. We then analyzed the given equation to determine the hyperbola's orientation and to extract the values of 'a' and 'b'. Using the relationship $c^2 = a^2 + b^2$, we calculated the distance 'c' from the center to each focus. Finally, we used this distance and the center coordinates to find the foci coordinates, which matched option A: (-20, 2) and (48, 2). This comprehensive approach highlights the importance of understanding the fundamental properties of hyperbolas and applying them systematically to solve problems. Mastering hyperbola concepts is not only crucial for academic success but also for various applications in engineering, physics, and other scientific fields. This journey through the hyperbola's equation and its foci underscores the beauty and precision of mathematical analysis.