Foci Of Hyperbola Calculation And Verification

The fascinating world of conic sections brings us to the hyperbola, a curve characterized by its unique properties and elegant equations. In mathematics, understanding hyperbolas is crucial not only for academic pursuits but also for various applications in physics, engineering, and even astronomy. At its core, a hyperbola is defined as the locus of points where the difference of the distances from two fixed points, known as foci, is constant. These foci play a pivotal role in determining the shape and orientation of the hyperbola. The general equation of a hyperbola centered at $(h, k)$ with a horizontal transverse axis is given by:

$$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

Here, $(h, k)$ represents the center of the hyperbola, $a$ is the distance from the center to each vertex along the transverse axis, and $b$ is related to the conjugate axis. The distance from the center to each focus is denoted by $c$, and it is related to $a$ and $b$ by the equation:

$$c^2 = a^2 + b^2$$

The foci are located at $(h ± c, k)$. This relationship is paramount in determining the foci of a hyperbola given its equation. The equation provided in the problem is:

$$\frac{(x-1)^2}{25} - \frac{(y+3)^2}{9} = 1$$

Our task is to verify whether the given foci, $(1 + \sqrt{34}, -3)$ and $(1 - \sqrt{34}, -3)$, are indeed the foci of this hyperbola. This involves identifying the center, $a$, $b$, calculating $c$, and then confirming the foci coordinates.

To rigorously verify the foci of the given hyperbola, we need to follow a systematic approach. This involves extracting the key parameters from the hyperbola's equation, calculating the focal distance, and then confirming the coordinates of the foci. Let's break down the process:

1. Identify the Center $(h, k)$: From the equation $\frac{(x-1)^2}{25} - \frac{(y+3)^2}{9} = 1$, the center of the hyperbola is readily identified as $(h, k) = (1, -3)$.

2. Determine $a^2$ and $b^2$: The denominators under the squared terms give us $a^2$ and $b^2$. Here, $a^2 = 25$ and $b^2 = 9$. Therefore, $a = \sqrt{25} = 5$ and $b = \sqrt{9} = 3$.

3. Calculate $c$ using $c^2 = a^2 + b^2$: The distance $c$ from the center to each focus is crucial. Using the relationship $c^2 = a^2 + b^2$, we have $c^2 = 25 + 9 = 34$. Thus, $c = \sqrt{34}$.

4. Find the Foci Coordinates: For a hyperbola with a horizontal transverse axis, the foci are located at $(h ± c, k)$. Substituting the values we have, the foci are:

   • $(1 + \sqrt{34}, -3)$
   • $(1 - \sqrt{34}, -3)$

5. Compare with the Given Foci: The foci we calculated, $(1 + \sqrt{34}, -3)$ and $(1 - \sqrt{34}, -3)$, exactly match the foci provided in the problem statement. This confirms that our calculations are correct and the given foci are indeed the foci of the hyperbola.

Understanding the parameters of a hyperbola is essential for grasping its geometry and behavior. Let's delve deeper into the significance of the center, $a$, $b$, and $c$, and how they collectively define the hyperbola's shape and orientation.

• Center $(h, k)$: The center is the midpoint of the segment connecting the foci and serves as the hyperbola's central point of symmetry. In our example, the center $(1, -3)$ anchors the hyperbola in the coordinate plane. All other parameters are defined relative to this point. Shifting the center changes the hyperbola's position without altering its shape.

• $a$ – Distance from Center to Vertices: The parameter $a$ represents the distance from the center to each vertex along the transverse axis. The vertices are the points where the hyperbola intersects its transverse axis. In our case, $a = 5$, indicating that the vertices are 5 units away from the center along the horizontal axis. The vertices play a crucial role in defining the hyperbola's opening and overall size.

• $b$ – Related to the Conjugate Axis: The parameter $b$ is associated with the conjugate axis, which is perpendicular to the transverse axis. While $b$ doesn't directly define points on the hyperbola, it significantly influences the hyperbola's shape, particularly the asymptotes. Here, $b = 3$. The ratio $b/a$ determines the steepness of the hyperbola's branches.

• $c$ – Distance from Center to Foci: The distance $c$ is the most critical parameter when identifying the foci. It measures the distance from the center to each focus. The relationship $c^2 = a^2 + b^2$ ties $c$ to $a$ and $b$, making it dependent on both the transverse and conjugate axes. In our example, $c = \sqrt{34}$, which dictates the position of the foci at $(1 ± \sqrt{34}, -3)$.

By understanding these parameters, we gain a comprehensive understanding of the hyperbola's structure. Each parameter contributes uniquely to the hyperbola's characteristics, enabling us to analyze and manipulate these curves effectively.

One of the defining features of a hyperbola is its asymptotes—lines that the hyperbola approaches as it extends to infinity. Asymptotes provide valuable insights into the hyperbola's long-term behavior and overall shape. The equations of the asymptotes for a hyperbola centered at $(h, k)$ with a horizontal transverse axis are:

$$y - k = ± \frac{b}{a}(x - h)$$

In our specific hyperbola, $\frac{(x-1)^2}{25} - \frac{(y+3)^2}{9} = 1$, the center is $(1, -3)$, $a = 5$, and $b = 3$. Plugging these values into the equations, we get the asymptotes:

$$y + 3 = ± \frac{3}{5}(x - 1)$$

These asymptotes act as guidelines for the hyperbola's branches, showing the direction in which the hyperbola extends indefinitely. The slopes of the asymptotes, $± \frac{b}{a}$, determine how rapidly the hyperbola diverges from its center. A larger $b/a$ ratio indicates steeper asymptotes and a wider hyperbola, while a smaller ratio results in flatter asymptotes and a narrower hyperbola.

To visualize the asymptotes, imagine drawing lines through the center of the hyperbola with slopes of $3/5$ and $-3/5$. The hyperbola's branches will approach these lines more and more closely as they extend away from the center, but they will never actually intersect them. The asymptotes, therefore, serve as a crucial tool in sketching and understanding the shape of the hyperbola.

Furthermore, the asymptotes are instrumental in analyzing the hyperbola's behavior in various applications. For instance, in physics, the paths of particles moving under certain forces can be modeled by hyperbolas, and the asymptotes describe the particles' trajectories as they move far from the force's origin. Similarly, in engineering, the design of certain structures and systems may involve hyperbolic shapes, and the asymptotes play a role in ensuring stability and performance.

In conclusion, through a detailed step-by-step analysis, we have successfully verified that the foci for the hyperbola $\frac{(x-1)^2}{25} - \frac{(y+3)^2}{9} = 1$ are indeed $(1 + \sqrt{34}, -3)$ and $(1 - \sqrt{34}, -3)$. We began by identifying the center of the hyperbola as $(1, -3)$ and determining the values of $a$ and $b$ from the equation. Using the relationship $c^2 = a^2 + b^2$, we calculated $c = \sqrt{34}$, which is the distance from the center to each focus. Applying the standard formula for the foci of a hyperbola with a horizontal transverse axis, we found the foci to be exactly as stated in the problem.

This exercise underscores the importance of understanding the fundamental properties of hyperbolas, including the roles of the center, vertices, foci, and asymptotes. Each parameter contributes uniquely to the hyperbola's shape and orientation, and a thorough grasp of these concepts is essential for solving problems and applying hyperbolas in real-world contexts. From astronomy to engineering, hyperbolas appear in diverse fields, making their study both intellectually rewarding and practically valuable.

By mastering the techniques for analyzing hyperbolas, such as identifying key parameters and calculating foci, we empower ourselves to tackle more complex mathematical challenges and appreciate the elegance and utility of conic sections. The journey through the world of hyperbolas is not only about memorizing formulas but also about developing a deep understanding of the underlying principles that govern these fascinating curves.

Therefore, the assertion that the foci for the hyperbola $\frac{(x-1)^2}{25} - \frac{(y+3)^2}{9} = 1$ are $(1 + \sqrt{34}, -3)$ and $(1 - \sqrt{34}, -3)$ is definitively true.