Solving Hyperbola Equation Finding B Value For Equation Y^2 8^2 - X^2 B^2

At the heart of conic sections lies the hyperbola, a fascinating curve defined by its unique geometric properties. To truly grasp the intricacies of the given equation and determine the value of $b$, we must first understand the fundamental characteristics of hyperbolas. A hyperbola is defined as the locus of points where the difference of the distances to two fixed points, called foci, remains constant. This definition leads to the hyperbola's distinctive two-branch shape, each curving away from the center.

The standard equation of a hyperbola centered at the origin with a vertical transverse axis is given by:

$$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$$

In this equation:

• $a$ represents the distance from the center to each vertex (the points where the hyperbola intersects its transverse axis).
• $b$ is related to the distance from the center to the co-vertices (the endpoints of 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 $c^2 = a^2 + b^2$.

In our problem, the given equation is $\frac{y^2}{8^2} - \frac{x^2}{b^2} = 1$, which is a hyperbola centered at the origin with a vertical transverse axis. We are also given that one of the foci is at $(0, -10)$. This crucial piece of information will help us determine the value of $b$. Analyzing the given equation, we can immediately identify that $a^2 = 8^2$, which means $a = 8$. Since the focus is at $(0, -10)$, the distance from the center $(0, 0)$ to the focus is $c = 10$. With the values of $a$ and $c$ known, we can use the relationship $c^2 = a^2 + b^2$ to solve for $b$. This foundational understanding of hyperbolas and their properties is paramount to tackling the problem at hand and arriving at the correct solution. Without a solid grasp of these principles, navigating the equation and the given information becomes a challenging endeavor. Therefore, we will delve deeper into the relationship between $a$, $b$, and $c$, and how they define the shape and position of the hyperbola.

Now, let's delve into the heart of the problem. The given equation, $\frac{y^2}{8^2} - \frac{x^2}{b^2} = 1$, immediately reveals that $a = 8$, as $a^2$ is the denominator under the $y^2$ term. The focus at $(0, -10)$ tells us that $c = 10$, representing the distance from the center to the focus. The key relationship that connects $a$, $b$, and $c$ in a hyperbola is $c^2 = a^2 + b^2$. This equation stems from the Pythagorean theorem and the geometry of the hyperbola, providing a crucial link between the distances involved.

Substituting the known values of $a$ and $c$ into this equation, we get:

$10^2 = 8^2 + b^2$

Simplifying, we have:

$100 = 64 + b^2$

Subtracting 64 from both sides, we isolate $b^2$:

$b^2 = 100 - 64$

$b^2 = 36$

Taking the square root of both sides, we find:

$b = 6$

Therefore, the value of $b$ is 6. This methodical approach, utilizing the fundamental equation of a hyperbola and the relationship between its key parameters, allows us to solve for the unknown value. This process highlights the importance of understanding the underlying principles of conic sections and their corresponding equations. Each term in the equation holds a specific meaning, and by carefully dissecting the given information, we can unlock the solution. The relationship $c^2 = a^2 + b^2$ is not just a formula; it is a geometric truth that governs the shape and dimensions of the hyperbola. The larger the value of $b$, the wider the hyperbola opens along its conjugate axis. Conversely, a smaller value of $b$ results in a narrower hyperbola. Understanding this interplay between $a$, $b$, and $c$ provides a deeper appreciation for the nature of hyperbolas.

Through the methodical application of the hyperbola equation and the relationship between its parameters, we have successfully determined the value of $b$. By substituting the known values of $a$ and $c$ into the equation $c^2 = a^2 + b^2$, we found that $b^2 = 36$, which leads to $b = 6$. Therefore, the correct answer is B. 6. This problem serves as a testament to the power of understanding fundamental mathematical principles and their applications.

In summary, we began by understanding the definition and properties of a hyperbola, including its standard equation and the relationship between $a$, $b$, and $c$. We then applied this knowledge to the given problem, extracting the values of $a$ and $c$ from the equation and the given focus. By substituting these values into the equation $c^2 = a^2 + b^2$, we were able to solve for $b$. This step-by-step approach not only led us to the correct answer but also reinforced our understanding of the underlying concepts. The ability to manipulate equations, apply formulas, and interpret geometric relationships is crucial in mathematics. This problem encapsulates these skills and demonstrates how they can be used to solve complex problems.