Hyperbola Center Focus And Directrix A Detailed Explanation
In the fascinating world of conic sections, hyperbolas stand out with their unique properties and elegant curves. Understanding the fundamental components of a hyperbola, such as its center, foci, and directrices, is crucial for grasping its geometric nature and applications. In this comprehensive exploration, we will delve into the specifics of a hyperbola centered at the origin, with a focus at (0, 20) and a corresponding directrix at y = -256/20. We will analyze the relationships between these elements, derive the equation of the hyperbola, and discuss the significance of its vertices.
Before we dive into the specifics of our problem, let's establish a solid foundation by defining what a hyperbola is. A hyperbola is a type of conic section formed by the intersection of a double cone with a plane. It is characterized by two separate branches that extend infinitely, each curving away from the other. Key components of a hyperbola include:
- Center: The midpoint between the two foci.
- Foci (plural of focus): Two fixed points that define the hyperbola. The difference in distances from any point on the hyperbola to the two foci is constant.
- Vertices: The points where the hyperbola intersects its major axis (the axis passing through the foci).
- Directrices: Two lines associated with the hyperbola. For any point on the hyperbola, the ratio of its distance to a focus and its distance to the corresponding directrix is a constant value greater than 1, called the eccentricity.
- Asymptotes: Two lines that the hyperbola approaches as it extends infinitely. These lines intersect at the center of the hyperbola.
Our specific problem involves a hyperbola centered at the origin (0, 0). We are given that one focus is located at (0, 20) and its associated directrix is the line y = -256/20. We aim to determine the equation of this hyperbola and the location of its vertices. This exploration will provide a practical application of the general properties of hyperbolas.
Determining the Orientation
Since the focus is located at (0, 20) on the y-axis and the center is at the origin, we know that the hyperbola has a vertical transverse axis. This means the hyperbola opens upwards and downwards, and its equation will have the form:
(y^2 / a^2) - (x^2 / b^2) = 1
where 'a' is the distance from the center to each vertex, and 'b' is related to the distance between the vertices of the conjugate axis.
Understanding the Focus and Directrix Relationship
For a hyperbola with a vertical transverse axis, the foci are located at (0, ±c), where 'c' is the distance from the center to each focus. The directrices are horizontal lines given by the equations y = ±a²/c. We are given that one focus is at (0, 20), so c = 20. The corresponding directrix is y = -256/20, which simplifies to y = -64/5. Using the relationship between the directrix and the parameters of the hyperbola, we have:
a^2 / c = 64/5
Since we know c = 20, we can substitute this value into the equation:
a^2 / 20 = 64/5
Multiplying both sides by 20, we get:
a^2 = (64/5) * 20 = 64 * 4 = 256
Taking the square root of both sides, we find:
a = 16
Finding the Vertices
The vertices of a hyperbola with a vertical transverse axis are located at (0, ±a). Since we found a = 16, the vertices of our hyperbola are at (0, 16) and (0, -16).
Determining the Value of b
To complete the equation of the hyperbola, we need to find the value of 'b'. We can use the relationship between a, b, and c in a hyperbola:
c^2 = a^2 + b^2
We know c = 20 and a = 16, so we can plug these values into the equation:
20^2 = 16^2 + b^2
400 = 256 + b^2
Subtracting 256 from both sides, we get:
b^2 = 400 - 256 = 144
Taking the square root of both sides, we find:
b = 12
Writing the Equation of the Hyperbola
Now that we have the values of a and b, we can write the equation of the hyperbola:
(y^2 / a^2) - (x^2 / b^2) = 1
Substituting a = 16 and b = 12, we get:
(y^2 / 256) - (x^2 / 144) = 1
This is the equation of the hyperbola centered at the origin with a focus at (0, 20) and a directrix at y = -256/20.
The vertices of the hyperbola are crucial points that define its shape and orientation. For a hyperbola centered at the origin with a vertical transverse axis, the vertices are located at (0, ±a). In our case, we determined that a = 16. Therefore, the vertices of the hyperbola are:
- (0, 16)
- (0, -16)
These points represent the locations where the hyperbola intersects its major axis, which in this case is the y-axis. They are the points on the hyperbola closest to the center and play a significant role in defining the hyperbola's branches.
In summary, we have analyzed a hyperbola centered at the origin with a focus at (0, 20) and a directrix at y = -256/20. We determined the following:
- The hyperbola has a vertical transverse axis.
- The distance from the center to each focus (c) is 20.
- The distance from the center to each vertex (a) is 16.
- The value of b is 12.
- The equation of the hyperbola is (y^2 / 256) - (x^2 / 144) = 1.
- The vertices of the hyperbola are (0, 16) and (0, -16).
Understanding the relationships between the center, foci, directrices, and vertices is essential for working with hyperbolas. By applying these concepts and the standard equation of a hyperbola, we can analyze and define these conic sections effectively. This exploration not only provides a solution to the specific problem but also reinforces the fundamental principles of hyperbolas. The ability to determine the equation and key features of a hyperbola from its given properties is a valuable skill in mathematics and its applications, such as physics, engineering, and astronomy. Mastering the concepts related to hyperbolas allows us to model and understand various phenomena in the world around us, from the trajectories of celestial bodies to the design of lenses and reflectors.
In conclusion, the hyperbola centered at the origin with a focus at (0, 20) and a directrix at y = -256/20 is a testament to the beauty and complexity of conic sections. Through careful analysis and application of fundamental principles, we have successfully derived its equation and identified its key features, including its vertices. This comprehensive exploration serves as a valuable resource for anyone seeking a deeper understanding of hyperbolas and their significance in mathematics and beyond. The knowledge gained from this analysis can be applied to a wide range of problems and real-world applications, further solidifying the importance of studying these fascinating curves.
