Hyperbola Equation Analysis Finding Center And Vertices

In the realm of analytic geometry, hyperbolas stand out as fascinating conic sections, distinguished by their unique shape and properties. Unlike ellipses, which are defined by the sum of distances to two fixed points (foci), hyperbolas are defined by the difference of these distances. This subtle difference leads to a dramatic change in the curve's appearance, resulting in two separate branches that extend infinitely. To fully grasp the essence of a hyperbola, we must delve into its key features: the center, vertices, foci, and asymptotes.

Consider the equation provided: $-\frac{(x+5)^2}{9^3}+\frac{(y-7)^2}{13^3}=1$. This equation represents a hyperbola in standard form, a format that reveals crucial information about the curve's position and orientation. Let's dissect this equation piece by piece to uncover the secrets it holds. The standard form equation serves as a roadmap, guiding us to pinpoint the hyperbola's center, which acts as the midpoint of the hyperbola. The center serves as the anchor point from which all other features are referenced. Additionally, we can identify the vertices. The vertices are the points where the hyperbola intersects its main axis. This axis depends on whether the hyperbola opens horizontally or vertically. Finally, we can understand the orientation, which dictates whether the hyperbola opens horizontally or vertically, a characteristic determined by the sign of the terms in the equation. By carefully examining the standard form equation, we unlock the hyperbola's fundamental characteristics, setting the stage for further exploration of its geometric intricacies.

The heart of a hyperbola lies in its center, the central point around which the curve is symmetrically balanced. In the given equation, $-\frac{(x+5)^2}{9^3}+\frac{(y-7)^2}{13^3}=1$, the center's coordinates are elegantly encoded within the equation's structure. By recognizing the standard form of a hyperbola equation, we can effortlessly extract this vital information. The standard form equation reveals that the center is located at the point (-5, 7). This is a foundational step in understanding the hyperbola's position on the coordinate plane.

Now, let's turn our attention to the hyperbola's orientation. Does it open horizontally, stretching along the x-axis, or vertically, extending along the y-axis? The answer lies in the signs preceding the terms in the equation. In our case, the term involving (y - 7)^2 is positive, while the term involving (x + 5)^2 is negative. This crucial difference indicates that the hyperbola opens vertically. The positive term always corresponds to the axis along which the hyperbola opens. Visualizing this, we can imagine the hyperbola's branches extending upwards and downwards from the center, creating a vertical orientation. Understanding the orientation is essential for determining the location of the vertices and foci, as well as for sketching an accurate representation of the hyperbola. Therefore, by analyzing the signs in the equation, we've successfully deciphered the hyperbola's vertical orientation, adding another layer to our understanding of its geometric properties.

The vertices of a hyperbola mark the extreme points on its main axis, the axis that passes through the center and the foci. Since we've established that our hyperbola opens vertically, the vertices will be located above and below the center. To pinpoint their exact coordinates, we need to consider the value under the positive term in the equation, which in our case is 13^3. This value, often denoted as a^2, plays a crucial role in determining the distance from the center to each vertex.

In our equation, a^2 = 13^3. Taking the square root of both sides, we find that a = √13^3 = 13√13. This value represents the distance from the center to each vertex along the vertical axis. Starting from the center (-5, 7), we move a distance of 13√13 units upwards and downwards to locate the vertices. This means the top vertex is located at (-5, 7 + 13√13), and the bottom vertex is located at (-5, 7 - 13√13). However, the question asks specifically for the bottom vertex. Therefore, the bottom vertex of the hyperbola is (-5, 7 - 13√13).

By carefully considering the orientation and the value of a, we've successfully located the vertices, the extreme points that define the hyperbola's vertical extent. These points, along with the center, provide a framework for visualizing and understanding the hyperbola's shape and position. Identifying the vertices is a key step in characterizing the hyperbola's geometry, paving the way for further exploration of its properties and applications.

In summary, for the hyperbola represented by the equation $-\frac{(x+5)^2}{9^3}+\frac{(y-7)^2}{13^3}=1$, we have determined the following:

• The center of the hyperbola is located at the point (-5, 7).
• The bottom vertex of the hyperbola is located at the point (-5, 7 - 13√13). This is because the hyperbola opens vertically, and the bottom vertex is the lowest point on the hyperbola.

These two points, the center and the bottom vertex, provide a fundamental understanding of the hyperbola's position and orientation in the coordinate plane. They serve as anchor points for further analysis, such as determining the foci, asymptotes, and overall shape of the hyperbola. By mastering the process of identifying these key features, we gain a deeper appreciation for the intricacies of hyperbolas and their role in various mathematical and scientific applications.