Hyperbola Equation Find Equation From Center Vertex And Focus

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In the realm of conic sections, hyperbolas stand out as fascinating curves with unique properties and equations. Understanding hyperbolas requires a grasp of their key components: the center, vertices, foci, and asymptotes. In this comprehensive analysis, we will delve into the equation of a hyperbola, focusing on how to determine the equation given specific parameters such as the center, vertex, and focus. Our primary goal is to identify which equation accurately represents a hyperbola with a center at (0, 0), a vertex at (-48, 0), and a focus at (50, 0). This exploration will not only provide a solution to the problem but also enhance your understanding of the fundamental principles governing hyperbolas.

Defining the Hyperbola

A hyperbola is defined as the set of all points in a plane such that the absolute difference of the distances between two fixed points, called the foci, is constant. This constant difference is a critical characteristic that distinguishes hyperbolas from other conic sections like ellipses and parabolas. The center of a hyperbola is the midpoint between the two foci, serving as the central point around which the hyperbola is symmetric. The vertices are the points on the hyperbola that are closest to the center; they lie on the transverse axis, which is the axis passing through the foci and the center. The asymptotes are lines that the hyperbola approaches as it extends towards infinity, providing a framework for the hyperbola's branches.

Key Components and Their Significance

  1. Center (h, k): The center is the reference point for the hyperbola. Its coordinates (h, k) determine the hyperbola's position in the coordinate plane. For a hyperbola centered at the origin (0, 0), the equations are simplified, making analysis more straightforward.
  2. Vertices (±a, 0) or (0, ±a): The vertices are the points where the hyperbola intersects its transverse axis. The distance from the center to a vertex is denoted as 'a'. The vertices play a crucial role in defining the shape and orientation of the hyperbola.
  3. Foci (±c, 0) or (0, ±c): The foci are the fixed points used in the definition of the hyperbola. The distance from the center to a focus is denoted as 'c'. The foci are always farther from the center than the vertices, and their position significantly influences the hyperbola's curvature.
  4. Asymptotes: Asymptotes are lines that the hyperbola approaches as it extends infinitely. They intersect at the center of the hyperbola and provide a guideline for sketching the hyperbola's branches. The equations of the asymptotes are determined by the values of 'a' and 'b', where 'b' is related to the conjugate axis (the axis perpendicular to the transverse axis).

Standard Equations of Hyperbolas

The standard form of the equation for a hyperbola depends on whether the transverse axis is horizontal or vertical. Understanding these standard forms is essential for identifying the hyperbola's key parameters and graphing it accurately.

Horizontal Transverse Axis

For a hyperbola with a horizontal transverse axis and center at (0, 0), the standard equation is:

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

In this equation:

  • 'a' is the distance from the center to each vertex along the x-axis.
  • 'b' is related to the distance along the conjugate axis (y-axis).
  • The vertices are located at (±a, 0).
  • The foci are located at (±c, 0), where c is related to a and b by the equation c^2 = a^2 + b^2.
  • The asymptotes are given by the equations y = ±(b/a)x.

Vertical Transverse Axis

For a hyperbola with a vertical transverse axis and center at (0, 0), the standard equation is:

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

In this equation:

  • 'a' is the distance from the center to each vertex along the y-axis.
  • 'b' is related to the distance along the conjugate axis (x-axis).
  • The vertices are located at (0, ±a).
  • The foci are located at (0, ±c), where c is related to a and b by the equation c^2 = a^2 + b^2.
  • The asymptotes are given by the equations y = ±(a/b)x.

Determining the Equation from Given Parameters

To determine the equation of a hyperbola, we need to identify the values of 'a', 'b', and the orientation of the transverse axis. The given parameters, such as the center, vertex, and focus, provide the necessary information to find these values. Let's apply this process to the given problem: a hyperbola with a center at (0, 0), a vertex at (-48, 0), and a focus at (50, 0).

Step-by-Step Analysis

  1. Identify the Center: The center is given as (0, 0), which simplifies our equations since we can use the standard forms directly.

  2. Determine the Orientation of the Transverse Axis: The vertex (-48, 0) and the focus (50, 0) lie on the x-axis. This indicates that the transverse axis is horizontal. Therefore, we will use the standard equation for a hyperbola with a horizontal transverse axis:

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

  3. Find the Value of 'a': The vertex is at (-48, 0), which means the distance from the center (0, 0) to the vertex is 48. Thus, a = 48.

  4. Find the Value of 'c': The focus is at (50, 0), so the distance from the center (0, 0) to the focus is 50. Thus, c = 50.

  5. Find the Value of 'b': We use the relationship c^2 = a^2 + b^2 to find b. We have c = 50 and a = 48, so:

    50^2 = 48^2 + b^2

    2500 = 2304 + b^2

    b^2 = 2500 - 2304

    b^2 = 196

    b = 14

  6. Write the Equation: Now that we have a = 48 and b = 14, we can write the equation of the hyperbola:

    (x^2 / 48^2) - (y^2 / 14^2) = 1

    (x^2 / 2304) - (y^2 / 196) = 1

Conclusion

In summary, we have successfully determined the equation of a hyperbola with a center at (0, 0), a vertex at (-48, 0), and a focus at (50, 0). By understanding the definitions and standard equations of hyperbolas, we were able to systematically derive the equation. The equation that represents this hyperbola is:

(x^2 / 2304) - (y^2 / 196) = 1

This process highlights the importance of recognizing the key parameters of a hyperbola and their relationship to its equation. The values of 'a', 'b', and 'c', along with the orientation of the transverse axis, are crucial in defining the hyperbola's shape and position in the coordinate plane. By mastering these concepts, you can confidently analyze and solve problems involving hyperbolas, enriching your understanding of conic sections and their applications in mathematics and beyond.

Which equation represents the hyperbola given its center at (0, 0), a vertex at (-48, 0), and a focus at (50, 0)?

Hyperbola Equation Find Equation from Center Vertex and Focus