Hyperbola Equation Finding Center Focus And Directrix

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In the captivating realm of conic sections, hyperbolas stand out with their distinctive two-branched shape. Understanding the properties and equations of hyperbolas is crucial for various applications in physics, engineering, and mathematics. In this comprehensive guide, we will delve into the process of determining the equation of a hyperbola given its center, focus, and directrix. This problem involves a hyperbola centered at the origin, with a focus at (0, 20) and a corresponding directrix defined by the line y = -256/20. We will meticulously walk through each step, ensuring a clear and thorough understanding of the underlying concepts.

1. Grasping the Fundamentals of Hyperbolas

To effectively tackle this problem, it's essential to have a solid grasp of the fundamental properties of hyperbolas. A hyperbola is defined as the set of all points where the difference of the distances to two fixed points, called the foci, is constant. This constant difference is typically denoted as 2a, where 'a' is the distance from the center of the hyperbola to each vertex. The center of the hyperbola is the midpoint of the segment connecting the two foci. The vertices are the points where the hyperbola intersects its principal axis, which is the line passing through the foci and the center. A directrix is a line associated with each focus, and the ratio of the distance from a point on the hyperbola to the focus and the distance to the directrix is a constant value called the eccentricity (e), where e > 1 for a hyperbola.

1.1. Key Parameters of a Hyperbola

Before diving into the specific problem, let's define the key parameters that characterize a hyperbola:

  • Center (h, k): The midpoint of the segment connecting the two foci. In our case, the center is given as the origin (0, 0).
  • Foci (h, k ± c): The two fixed points used in the definition of the hyperbola. The distance from the center to each focus is denoted as 'c'. We are given one focus at (0, 20), which implies c = 20 since the center is at the origin.
  • Vertices (h, k ± a): The points where the hyperbola intersects its principal axis. The distance from the center to each vertex is denoted as 'a'.
  • Directrices y = k ± a/e: Lines associated with each focus. The distance from the center to each directrix is a/e, where 'e' is the eccentricity. We are given the equation of one directrix as y = -256/20.
  • Eccentricity (e): A constant value greater than 1 that determines the shape of the hyperbola. It is defined as the ratio of the distance from the center to a focus (c) to the distance from the center to a vertex (a), i.e., e = c/a.
  • Asymptotes: Lines that the hyperbola approaches as it extends infinitely. The equations of the asymptotes depend on the orientation of the hyperbola.

1.2. Standard Equation of a Hyperbola

The standard equation of a hyperbola depends on whether the transverse axis (the axis passing through the foci and vertices) is horizontal or vertical. Since the focus (0, 20) lies on the y-axis and the center is at the origin, the transverse axis is vertical. The standard equation for a hyperbola with a vertical transverse axis and center at the origin is:

y^2/a^2 - x^2/b^2 = 1

where:

  • 'a' is the distance from the center to each vertex.
  • 'b' is related to the distance between the asymptotes and is determined by the equation b^2 = c^2 - a^2.
  • 'c' is the distance from the center to each focus.

2. Decoding the Given Information

Now, let's analyze the information provided in the problem statement:

  • Center: The center of the hyperbola is located at the origin (0, 0). This simplifies our equation since h = 0 and k = 0.
  • Focus: One focus is located at (0, 20). This tells us that c = 20, as the distance from the center (0, 0) to the focus (0, 20) is 20 units.
  • Directrix: The associated directrix is represented by the line y = -256/20. Since the transverse axis is vertical, the directrices are horizontal lines of the form y = ± a/e. The directrix y = -256/20 corresponds to the directrix below the center, so we have a/e = 256/20.

3. Unraveling the Equation: A Step-by-Step Approach

Our goal is to find the equation of the hyperbola in the standard form: y2/a2 - x2/b2 = 1. To achieve this, we need to determine the values of 'a' and 'b'. We already know that c = 20. Let's use the information about the directrix to find 'a' and 'e'.

3.1. Finding 'a' and 'e'

We have two crucial pieces of information:

  1. a/e = 256/20
  2. e = c/a = 20/a

We can substitute the second equation into the first equation to eliminate 'e':

a / (20/a) = 256/20

Simplifying this equation, we get:

a^2 / 20 = 256/20

Multiplying both sides by 20, we obtain:

a^2 = 256

Taking the square root of both sides, we find:

a = 16

Now that we have 'a', we can find 'e' using the equation e = c/a:

e = 20/16 = 5/4

3.2. Determining 'b'

To find 'b', we use the relationship b^2 = c^2 - a^2:

b^2 = 20^2 - 16^2
b^2 = 400 - 256
b^2 = 144

Taking the square root of both sides, we get:

b = 12

4. Constructing the Hyperbola Equation

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
y^2/16^2 - x^2/12^2 = 1
y^2/256 - x^2/144 = 1

Therefore, the equation of the hyperbola is:

y^2/256 - x^2/144 = 1

5. Visualizing the Hyperbola

To further solidify our understanding, let's visualize the hyperbola. We know the center is at (0, 0), the vertices are at (0, ±16), and the foci are at (0, ±20). The asymptotes can be found using the equation y = ±(a/b)x, which in this case is y = ±(16/12)x or y = ±(4/3)x. By plotting these points and lines, we can sketch the hyperbola and gain a visual representation of its shape and characteristics.

6. Conclusion: Mastering Hyperbola Equations

In this detailed exploration, we successfully determined the equation of a hyperbola given its center, focus, and directrix. We meticulously dissected the problem, starting with the fundamental properties of hyperbolas and progressing through each step of the solution. By understanding the relationships between the key parameters – center, foci, vertices, directrices, eccentricity, and asymptotes – we were able to construct the equation of the hyperbola. This comprehensive guide serves as a valuable resource for anyone seeking to master the intricacies of hyperbola equations and their applications in various fields.

Mastering hyperbolas requires a blend of understanding the fundamental definitions, key parameters, and their relationships. By applying this knowledge systematically, one can confidently tackle a variety of problems involving hyperbolas. Remember, the key is to break down the problem into manageable steps, carefully analyze the given information, and apply the appropriate formulas and relationships. With practice and a solid understanding of the concepts, you can confidently navigate the world of hyperbolas and their fascinating properties.

Delving into the realm of conic sections, the hyperbola presents a fascinating geometric figure with distinctive properties and applications. This article serves as a comprehensive guide to understanding and deriving the equation of a hyperbola, particularly when given its center, focus, and directrix. Our specific case involves a hyperbola centered at the origin, with a focus located at (0, 20) and its corresponding directrix represented by the line y = -256/20. We will meticulously dissect the problem, elucidating each step and ensuring a thorough comprehension of the underlying mathematical principles. This exploration aims to empower readers with the knowledge and skills to confidently tackle similar problems and appreciate the elegance of hyperbolic geometry.

1. The Essence of Hyperbolas: Unveiling the Geometric Definition

At its core, a hyperbola is defined as the locus of points where the absolute difference of the distances to two fixed points, known as the foci, remains constant. This constant difference is conventionally denoted as 2a, where 'a' represents the distance from the center of the hyperbola to each of its vertices. The center of the hyperbola is the midpoint of the line segment connecting the two foci, effectively serving as the hyperbola's central point of symmetry. The vertices are the points where the hyperbola intersects its principal axis, which is the line passing through the foci and the center. A directrix is a line associated with each focus, and the ratio of the distance from any point on the hyperbola to the focus and its distance to the corresponding directrix is a constant value called the eccentricity (e), where e > 1 for a hyperbola. This eccentricity plays a pivotal role in shaping the hyperbola, dictating its degree of curvature and the separation between its branches.

1.1. Key Attributes of a Hyperbola: A Comprehensive Overview

To effectively analyze and manipulate hyperbolas, it's crucial to grasp the significance of their key attributes. These parameters not only define the hyperbola's shape and position but also serve as the building blocks for deriving its equation. Let's delve into these attributes:

  • Center (h, k): The central point of the hyperbola, serving as its midpoint of symmetry. In our case, the center is explicitly given as the origin (0, 0), simplifying our calculations.
  • Foci (h, k ± c): The two fixed points that define the hyperbola. The distance from the center to each focus is denoted by 'c'. In our problem, one focus is located at (0, 20), implying that c = 20 since the center is at the origin and the transverse axis is vertical.
  • Vertices (h, k ± a): The points where the hyperbola intersects its principal axis. The distance from the center to each vertex is denoted by 'a'. This parameter is crucial in determining the hyperbola's overall size and shape.
  • Directrices (y = k ± a/e): Lines associated with each focus, dictating the eccentricity and, consequently, the shape of the hyperbola. The distance from the center to each directrix is a/e, where 'e' represents the eccentricity. We are provided with the equation of one directrix as y = -256/20, a key piece of information for unraveling the hyperbola's equation.
  • Eccentricity (e): A dimensionless quantity greater than 1 that governs the hyperbola's shape. It is defined as the ratio of the distance from the center to a focus (c) to the distance from the center to a vertex (a), expressed as e = c/a. A higher eccentricity signifies a more elongated hyperbola.
  • Asymptotes: Lines that the hyperbola approaches asymptotically as it extends infinitely. These lines provide a visual guide to the hyperbola's long-term behavior and are instrumental in sketching its graph. The equations of the asymptotes depend on the orientation of the hyperbola and its parameters.

1.2. The Standard Equation: A Mathematical Representation

The standard equation of a hyperbola is dictated by the orientation of its transverse axis, which is the axis passing through the foci and vertices. Since the given focus (0, 20) lies on the y-axis and the center is at the origin, the transverse axis is vertical. This dictates the form of the standard equation as:

y^2/a^2 - x^2/b^2 = 1

where:

  • 'a' represents the distance from the center to each vertex, a crucial parameter in defining the hyperbola's vertical extent.
  • 'b' is intrinsically linked to the distance between the asymptotes and is mathematically defined by the relationship b^2 = c^2 - a^2. This parameter governs the hyperbola's horizontal spread.
  • 'c' denotes the distance from the center to each focus, a fundamental parameter given in the problem statement.

2. Decoding the Given Information: Extracting Key Clues

To effectively construct the hyperbola's equation, we must meticulously analyze the information provided in the problem statement. This involves extracting key clues and translating them into mathematical parameters:

  • Center: The hyperbola's center is explicitly located at the origin (0, 0). This simplifies our equation since h = 0 and k = 0, eliminating translation terms.
  • Focus: One focus is positioned at (0, 20). This directly reveals that c = 20, as the distance from the center (0, 0) to the focus (0, 20) is precisely 20 units. This parameter anchors the hyperbola's vertical orientation and scale.
  • Directrix: The associated directrix is defined by the line y = -256/20. Since the transverse axis is vertical, the directrices are horizontal lines with the general form y = ± a/e. The given directrix y = -256/20 corresponds to the directrix positioned below the center, leading to the crucial relationship a/e = 256/20. This equation serves as a cornerstone in our quest to determine 'a' and 'e'.

3. Unraveling the Equation: A Step-by-Step Derivation

Our primary objective is to derive the equation of the hyperbola in its standard form: y2/a2 - x2/b2 = 1. This necessitates determining the values of 'a' and 'b', the key parameters that dictate the hyperbola's shape and dimensions. We already have c = 20, a crucial piece of the puzzle. We will now leverage the information about the directrix to unveil 'a' and 'e', paving the way for calculating 'b'.

3.1. Determining 'a' and 'e': A Mathematical Dance

We possess two pivotal equations:

  1. a/e = 256/20
  2. e = c/a = 20/a

The elegance of mathematics lies in its ability to manipulate equations to extract hidden information. We can strategically substitute the second equation into the first, effectively eliminating 'e' and isolating 'a':

a / (20/a) = 256/20

Simplifying this equation through algebraic manipulation, we arrive at:

a^2 / 20 = 256/20

Multiplying both sides by 20, we gracefully isolate a^2:

a^2 = 256

Taking the square root of both sides, we unveil the value of 'a':

a = 16

With 'a' in our grasp, we can now readily calculate 'e' using the equation e = c/a:

e = 20/16 = 5/4

3.2. Calculating 'b': Bridging the Parameters

To complete the equation, we need to determine 'b'. This is achieved through the fundamental relationship b^2 = c^2 - a^2, which connects 'b' to the known parameters 'c' and 'a':

b^2 = 20^2 - 16^2
b^2 = 400 - 256
b^2 = 144

Taking the square root of both sides, we obtain the value of 'b':

b = 12

4. Constructing the Hyperbola's Equation: The Grand Finale

With the values of 'a' and 'b' firmly in hand, we can now confidently construct the equation of the hyperbola, fulfilling our initial objective:

y^2/a^2 - x^2/b^2 = 1
y^2/16^2 - x^2/12^2 = 1
y^2/256 - x^2/144 = 1

Therefore, the equation of the hyperbola, elegantly capturing its geometric essence, is:

y^2/256 - x^2/144 = 1

5. Visualizing the Hyperbola: A Geometric Perspective

To deepen our understanding and solidify our grasp of the hyperbola, let's visualize its key features. We know that the center is located at (0, 0), providing the anchor point. The vertices, the points where the hyperbola intersects its transverse axis, are positioned at (0, ±16), defining the hyperbola's vertical extent. The foci, the defining points of the hyperbola, reside at (0, ±20), further emphasizing its vertical orientation. The asymptotes, those guiding lines that the hyperbola approaches at infinity, can be determined using the equation y = ±(a/b)x, which in our case simplifies to y = ±(16/12)x or y = ±(4/3)x. By plotting these points and lines, we can sketch the hyperbola, gaining a visual appreciation for its distinctive two-branched shape and its relationship to its center, foci, vertices, and asymptotes. This visualization serves as a powerful tool for reinforcing our understanding and fostering a deeper connection with the geometry of hyperbolas.

6. Conclusion: Mastering Hyperbolic Equations

In this comprehensive exploration, we have successfully navigated the intricacies of determining the equation of a hyperbola, armed with the knowledge of its center, focus, and directrix. We meticulously dissected the problem, commencing with the fundamental definition of a hyperbola and progressing through each step of the derivation with clarity and precision. By understanding the interrelationships between the key parameters – center, foci, vertices, directrices, eccentricity, and asymptotes – we were empowered to construct the hyperbola's equation with confidence. This journey underscores the power of geometric reasoning and algebraic manipulation in unraveling the mysteries of conic sections. This guide serves as a valuable resource for anyone seeking to master the art of hyperbola equations and their diverse applications in mathematics, physics, and engineering.

The journey to mastering hyperbolas is paved with a solid foundation in their definitions, key parameters, and interrelationships. By systematically applying this knowledge, one can confidently confront a multitude of problems involving hyperbolas. The key lies in breaking down complex problems into manageable steps, carefully extracting information from the given conditions, and applying the appropriate formulas and relationships. With persistent practice and a deep understanding of the underlying concepts, you can confidently navigate the world of hyperbolas, appreciating their elegance and harnessing their power in various domains.

Let's explore how to determine the equation of a hyperbola when given the location of its center, one focus, and its associated directrix. This is a classic problem in conic sections that combines geometric intuition with algebraic manipulation. In this particular case, we're given that the center of the hyperbola is at the origin (0,0), one focus is at (0,20), and the corresponding directrix is the line y = -256/20. We'll walk through the steps involved in finding the equation of this hyperbola, emphasizing the underlying principles and the connections between different parameters. Understanding this process will provide a solid foundation for tackling other hyperbola-related problems.

1. Understanding the Hyperbola's Definition and Key Properties

Before we dive into the calculations, it's crucial to have a firm grasp of the definition of a hyperbola and its key properties. A hyperbola is defined as the set of all points in a plane such that the absolute difference of the distances from two fixed points (called the foci) is constant. This constant difference is usually denoted as 2a, where 'a' is a parameter related to the hyperbola's shape. The center of the hyperbola is the midpoint of the segment connecting the two foci. The vertices are the points where the hyperbola intersects the line passing through the foci (this line is called the transverse axis). Each focus has an associated line called a directrix, and the ratio of the distance from any point on the hyperbola to the focus to its distance to the directrix is a constant greater than 1, called the eccentricity (denoted by 'e'). This eccentricity determines the