Equation Of A Hyperbola Centered At Origin With Vertex At (0,-8) And Focus At (0,10)

Understanding hyperbolas is crucial in analytic geometry, particularly when dealing with conic sections. This article dives deep into identifying the correct equation for a hyperbola given its center, vertex, and focus. We'll explore the properties of hyperbolas, the standard forms of their equations, and how to derive the equation from the provided information. Let's embark on this mathematical journey to master the intricacies of hyperbolas.

Understanding Hyperbolas

Hyperbolas are fascinating curves defined as the set of all points where the difference of the distances to two fixed points (foci) is constant. This definition is critical in understanding the properties and equations of hyperbolas. Before diving into the specific problem, let's solidify our understanding of the key characteristics of a hyperbola.

A hyperbola consists of two symmetrical branches that open away from each other. The center of the hyperbola is the midpoint between the two foci. The vertices are the points where the hyperbola intersects its principal axis, which is the line passing through the foci. The distance from the center to a vertex is denoted by a, and the distance from the center to a focus is denoted by c. Another important parameter is b, which is related to a and c by the equation c² = a² + b². The values of a and b determine the shape and orientation of the hyperbola. The relationship between a, b, and c is fundamental to determining the equation of a hyperbola. Understanding these parameters is essential for solving problems involving hyperbolas.

The orientation of a hyperbola depends on whether the transverse axis (the axis passing through the vertices and foci) is horizontal or vertical. If the transverse axis is horizontal, the standard form of the hyperbola's equation is ${(x^2/a^2) - (y^2/b^2) = 1}$. If the transverse axis is vertical, the standard form is ${(y^2/a^2) - (x^2/b^2) = 1}$. The key difference lies in which term, x² or y², comes first in the equation. This dictates whether the hyperbola opens horizontally or vertically. Recognizing this distinction is crucial for correctly formulating the hyperbola's equation based on the given information.

Analyzing the Given Information

To determine the correct equation for the hyperbola, let's analyze the given information meticulously. We are told that the hyperbola is centered at the origin (0,0), which simplifies our task considerably. The vertex is given as (0,-8), and the focus is at (0,10). Since both the vertex and the focus lie on the y-axis, we can conclude that the hyperbola has a vertical transverse axis. This is a crucial observation that will guide us in selecting the appropriate standard form of the equation.

Knowing that the hyperbola has a vertical transverse axis means that its equation will be of the form ${(y^2/a^2) - (x^2/b^2) = 1}$. This form is different from the one for a hyperbola with a horizontal transverse axis, where the x² term comes first. The location of the vertex and focus relative to the center helps determine the values of a and c. The distance from the center (0,0) to the vertex (0,-8) gives us the value of a, and the distance from the center (0,0) to the focus (0,10) gives us the value of c. These distances are essential for calculating a and c, which are then used to find b.

The distance from the center to the vertex is a, so in this case, a = |-8 - 0| = 8. Similarly, the distance from the center to the focus is c, so c = |10 - 0| = 10. Now that we have a and c, we can use the relationship c² = a² + b² to find b². Plugging in the values, we get 10² = 8² + b², which simplifies to 100 = 64 + b². Solving for b², we find b² = 100 - 64 = 36. This value is critical for completing the equation of the hyperbola. With a² and b² determined, we can construct the final equation.

Deriving the Equation

Now that we have identified the orientation of the hyperbola and calculated the values of a and b, we can derive the equation. As established earlier, since the hyperbola has a vertical transverse axis, its equation will be of the form ${(y^2/a^2) - (x^2/b^2) = 1}$. We found that a = 8, so a² = 64. We also found that b² = 36. Plugging these values into the standard form equation, we get ${(y^2/64) - (x^2/36) = 1}$. This equation represents the hyperbola centered at the origin with the given vertex and focus.

By substituting the calculated values into the standard form, we have successfully derived the equation that represents the hyperbola. This process involves a clear understanding of the hyperbola's properties and the relationship between its parameters. The equation ${(y^2/64) - (x^2/36) = 1}$ precisely describes the hyperbola with the specified characteristics. It’s important to note how each piece of information, from the center to the focus and vertex, plays a crucial role in determining the final equation.

Evaluating the Options

With the equation derived, we can now evaluate the given options to identify the correct one. The options provided are:

1. ${(x^2/64) - (y^2/36) = 1}$
2. ${(x^2/64) - (y^2/100) = 1}$
3. ${(y^2/64) - (x^2/36) = 1}$
4. ${(y^2/100) - (x^2/64) = 1}$

Comparing these options with the equation we derived, ${(y^2/64) - (x^2/36) = 1}$, we can clearly see that option 3 is the correct answer. This option matches our derived equation exactly, confirming that it represents the hyperbola centered at the origin with a vertex at (0,-8) and a focus at (0,10).

The other options can be eliminated because they do not match the standard form we derived. Option 1 represents a hyperbola with a horizontal transverse axis, which is not the case here. Option 2 also represents a hyperbola with a horizontal transverse axis and incorrect values for a and b. Option 4 has the correct orientation but incorrect values for a and b. Therefore, only option 3 accurately represents the hyperbola described in the problem statement.

Conclusion

In conclusion, the equation that represents a hyperbola centered at the origin with a vertex at (0,-8) and a focus at (0,10) is (y²/64) - (x²/36) = 1. This was determined by understanding the properties of hyperbolas, identifying the orientation of the transverse axis, calculating the values of a and b, and applying the standard form equation for a hyperbola with a vertical transverse axis. This process demonstrates a comprehensive approach to solving problems involving conic sections.

By carefully analyzing the given information and applying the principles of hyperbola equations, we were able to systematically derive the correct answer. This detailed explanation not only provides the solution but also enhances understanding of the underlying concepts. Mastering these concepts is crucial for success in analytic geometry and related fields. Understanding the relationship between the parameters of a hyperbola and its equation is key to solving these types of problems efficiently and accurately.