Finding Directrices Of A Hyperbola Centered At The Origin

by ADMIN 58 views

Hyperbolas, fascinating conic sections, hold a prominent place in mathematics and physics. Understanding their properties is crucial for solving various problems related to curves and their applications. This article delves into the specifics of a hyperbola centered at the origin, focusing on how to determine the equations of its directrices given the vertex and focus. We will walk through the process step-by-step, ensuring a comprehensive understanding of the underlying concepts.

Defining Hyperbolas and Their Key Components

Before we dive into the problem, let's establish a clear understanding of what a hyperbola is and its essential components. 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 foci (plural of focus), is constant. This constant difference is equal to the length of the transverse axis, which is the distance between the two vertices. The vertices are the points where the hyperbola intersects its main axis of symmetry.

Other key components of a hyperbola include:

  • Center: The midpoint of the segment connecting the two foci (or the two vertices).
  • Conjugate Axis: The axis perpendicular to the transverse axis, passing through the center.
  • Asymptotes: Two lines that the hyperbola approaches as it extends to infinity. These lines intersect at the center of the hyperbola.
  • Directrices: Two lines perpendicular to the transverse axis, located at a certain distance from the center. The distance between the center and each directrix is related to the hyperbola's eccentricity.
  • Eccentricity (e): A measure of how "stretched" the hyperbola is. It is defined as the ratio of the distance between the focus and the center to the distance between the vertex and the center. For hyperbolas, e > 1.

Hyperbolas Centered at the Origin: Standard Equations

When a hyperbola is centered at the origin (0, 0) of the coordinate plane, its equation takes one of two standard forms, depending on whether the transverse axis is horizontal or vertical:

  1. Horizontal Transverse Axis: The equation is given by:

    x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

    where:

    • a is the distance from the center to each vertex.
    • b is related to the distance from the center to the co-vertices (the endpoints of the conjugate axis).
    • The vertices are located at (±a, 0).
    • The foci are located at (±c, 0), where c^2 = a^2 + b^2.
    • The equations of the directrices are x = ±a²/c.

  2. Vertical Transverse Axis: The equation is given by:

    y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

    where:

    • a is the distance from the center to each vertex.
    • b is related to the distance from the center to the co-vertices.
    • The vertices are located at (0, ±a).
    • The foci are located at (0, ±c), where c^2 = a^2 + b^2.
    • The equations of the directrices are y = ±a²/c.

Problem Statement: A Hyperbola with Vertex at (3, 0) and Focus at (9, 0)

Now, let's tackle the specific problem. We are given a hyperbola centered at the origin with a vertex at (3, 0) and a focus at (9, 0). Our goal is to determine the equations of the directrices. To achieve this, we will follow these steps:

  1. Identify the Orientation: Determine whether the transverse axis is horizontal or vertical based on the location of the vertex and focus.
  2. Determine 'a' and 'c': Find the values of a (distance from center to vertex) and c (distance from center to focus).
  3. Calculate 'b': Use the relationship c^2 = a^2 + b^2 to find the value of b.
  4. Find a²/c: Calculate the value of a²/c, which is crucial for determining the directrices.
  5. Write the Equations of the Directrices: Based on the orientation of the hyperbola, write the equations of the directrices using the calculated value of a²/c.

Step-by-Step Solution

1. Identify the Orientation

The vertex is at (3, 0) and the focus is at (9, 0). Both points lie on the x-axis. Since the center is at the origin (0, 0), the transverse axis is horizontal. This means the equation of the hyperbola will be in the form $\frac{x2}{a2} - \frac{y2}{b2} = 1$.

2. Determine 'a' and 'c'

  • a is the distance from the center (0, 0) to the vertex (3, 0), so a = 3.
  • c is the distance from the center (0, 0) to the focus (9, 0), so c = 9.

3. Calculate 'b'

We use the relationship c^2 = a^2 + b^2 to find b. Plugging in the values of a and c, we get:

92=32+b29^2 = 3^2 + b^2

81=9+b281 = 9 + b^2

b2=72b^2 = 72

b=72=62b = \sqrt{72} = 6\sqrt{2}

4. Find a²/c

Now, we calculate a²/c:

a2c=329=99=1\frac{a^2}{c} = \frac{3^2}{9} = \frac{9}{9} = 1

5. Write the Equations of the Directrices

Since the transverse axis is horizontal, the equations of the directrices are of the form x = ±a²/c. We found that a²/c = 1, so the equations of the directrices are:

x=±1x = ±1

Therefore, the equations of the directrices are x = 1 and x = -1.

Conclusion: Mastering Hyperbola Equations

In this article, we successfully determined the equations of the directrices for a hyperbola centered at the origin with a vertex at (3, 0) and a focus at (9, 0). We accomplished this by understanding the fundamental properties of hyperbolas, identifying the orientation of the transverse axis, calculating the key parameters a, b, and c, and finally applying the formula for the directrices. This step-by-step approach provides a solid framework for tackling similar problems involving hyperbolas. Understanding the relationships between the center, vertices, foci, and directrices is essential for mastering the geometry of conic sections. Practice and application of these concepts will further solidify your understanding and problem-solving skills in this area of mathematics.

By carefully analyzing the given information and applying the standard formulas, we can confidently solve problems related to hyperbolas and their directrices. Remember to focus on the key properties and relationships to ensure accurate and efficient problem-solving. The world of conic sections offers many fascinating challenges, and a solid understanding of hyperbolas is a valuable asset in your mathematical journey.

This detailed exploration should equip you with the knowledge and skills to confidently approach and solve problems involving hyperbolas centered at the origin and their directrices. Remember to review the concepts, practice various examples, and build a strong foundation in conic sections for future mathematical endeavors.