Glossary

10.2 The Hyperbola

3 min read•june 25, 2024

Hyperbolas are fascinating conic sections with two infinite branches. They're formed when a plane intersects a perpendicular to its axis. Understanding their key points and equations is crucial for graphing and solving real-world problems.

Hyperbolas have practical applications in fields like astronomy, physics, and engineering. They're used in navigation systems, particle physics, and even satellite dish design. Mastering concepts opens doors to exciting scientific and technological advancements.

Hyperbola Fundamentals

Key points of hyperbolas

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  • formed by intersection of double cone and plane perpendicular to cone's axis
  • Two branches extend infinitely in opposite directions
  • is midpoint of line segment connecting
  • Vertices are points where hyperbola intersects
    • Transverse axis passes through center with endpoints at vertices
  • are two fixed points on transverse axis, equidistant from center
    • Distance from any point on hyperbola to foci has constant difference equal to distance between vertices
  • are straight lines that the hyperbola approaches but never intersects

Standard-form equations for hyperbolas

  • depends on orientation of transverse axis
    • (opens left and right): (x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
    • (opens up and down): (y−k)2a2−(x−h)2b2=1\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1
  • represents center of hyperbola
  • is distance from center to vertex on transverse axis
  • is distance from center to co-vertex on
    • Conjugate axis is perpendicular to transverse axis with endpoints at

Hyperbolas centered at origin

  • When center is at (0,0)(0, 0), standard form equations simplify to:
    • Horizontal transverse axis: x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
    • Vertical transverse axis: y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
  • Steps to sketch hyperbola centered at origin:
    1. Identify transverse axis direction (horizontal or vertical)
    2. Plot vertices on transverse axis at (±a,0)(\pm a, 0) or (0,±a)(0, \pm a)
    3. Plot co-vertices on conjugate axis at (0,±b)(0, \pm b) or (±b,0)(\pm b, 0)
    4. Draw hyperbola branches passing through vertices and extending infinitely

Advanced Hyperbola Concepts

Non-origin centered hyperbolas

  • For non-origin centers, use standard form equations with (h,k)(h, k)
  • Hyperbola is shifted horizontally by hh units and vertically by kk units from origin
  • Steps to graph non-origin centered hyperbola:
    1. Plot center (h,k)(h, k) on coordinate plane
    2. Determine transverse axis direction and values of aa and bb
    3. Plot vertices and co-vertices relative to center
    4. Draw hyperbola branches passing through vertices and extending infinitely

Real-world applications of hyperbolas

  • Applications in astronomy, physics, and engineering
  • Examples:
    • Locating object position using time difference of arrival in navigation systems ()
    • Describing particle path in gravitational or electrostatic field
    • Modeling reflective properties of satellite dishes and telescopes
  • Steps to solve real-world hyperbola problems:
    1. Identify given information and question to be answered
    2. Determine appropriate hyperbola equation based on context
    3. Substitute given values into equation and solve for unknown variable
    4. Interpret result in context of problem

Additional Hyperbola Properties

  • (e) measures how much the hyperbola deviates from a circular shape
  • is a line perpendicular to the transverse axis, used to define the hyperbola
  • is a chord of the hyperbola perpendicular to the transverse axis and passing through a focus

Key Terms to Review (21)

$(h, k)$: $(h, k)$ is the center point of a hyperbola, which is the point where the two branches of the hyperbola intersect. This point represents the coordinates that define the location of the hyperbola on the coordinate plane.
$a$: $a$ is a variable that represents a specific value or constant in mathematical expressions and equations. In the context of the hyperbola, $a$ is a parameter that defines the shape and size of the hyperbola.
$b$: $b$ is a variable that represents one of the key parameters in the equation of a hyperbola, a type of conic section. The hyperbola is a curve that is defined by its center, major axis, minor axis, and the values of $a$ and $b$, which determine the shape and orientation of the curve.
Asymptotes: An asymptote is a straight line that a curve approaches but never touches. It provides important information about the behavior and characteristics of a function as it approaches its limits.
Center: The center of a geometric shape, such as a circle or an ellipse, is the point that is equidistant from all points on the shape's perimeter. It is the point around which the shape is symmetrical and balanced.
Co-vertices: Co-vertices refer to the points on the x-axis where an ellipse or hyperbola intersects. These points are equidistant from the center of the curve and represent the maximum and minimum values of the curve along the x-axis.
Conic Section: A conic section is a two-dimensional shape that is formed by the intersection of a plane and a three-dimensional cone. These shapes include the ellipse, hyperbola, and parabola, which are fundamental in the study of analytic geometry and have numerous applications in various fields.
Conjugate Axis: The conjugate axis of a hyperbola is the line segment that passes through the center of the hyperbola and is perpendicular to the transverse axis. It represents the shorter of the two principal axes of the hyperbola.
Directrix: The directrix is a key concept in the study of conic sections, which are the shapes formed by the intersection of a plane and a cone. It is a fixed line that, along with the focus, defines the shape and properties of an ellipse, hyperbola, or parabola.
Double Cone: A double cone is a three-dimensional geometric shape that consists of two cones joined at their bases, forming a symmetrical figure with two pointed ends. This shape is particularly relevant in the context of the hyperbola, a conic section that can be visualized as the intersection of a double cone and a plane.
Eccentricity: Eccentricity is a measure of the shape and elongation of a conic section, such as an ellipse, hyperbola, or parabola. It describes how much the shape deviates from a perfect circle, with a value between 0 and 1 indicating the degree of elongation or flattening of the curve.
Foci: The foci of a conic section, such as an ellipse or hyperbola, are the two fixed points around which the curve is defined. They are the points of focus that determine the shape and properties of the conic section.
Horizontal Transverse Axis: The horizontal transverse axis, also known as the major axis, is a key component of the hyperbola, a conic section. It represents the longest distance across the hyperbola, running perpendicular to the vertical axis and passing through the center of the curve.
Hyperbola: A hyperbola is a type of conic section, which is a two-dimensional curve formed by the intersection of a plane and a cone. It is characterized by a distinctive shape that resembles two mirrored parabolas, with a center and two distinct branches that extend infinitely in opposite directions.
Latus Rectum: The latus rectum is a line segment that passes through the focus of a conic section and is perpendicular to the major axis. It is an important geometric property that helps characterize the shape and size of ellipses, hyperbolas, and parabolas.
Origin: The origin is a fundamental point of reference in coordinate systems, representing the intersection of the x-axis and y-axis where both values are zero. It serves as the starting point for measuring and describing the positions of points in a plane or space.
Standard Form: Standard form is a specific way of representing an equation or function that provides a clear and organized structure, making it easier to analyze and work with the mathematical expression. This form is particularly relevant in the context of linear functions, quadratic functions, and conic sections such as the ellipse, hyperbola, and parabola.
TDOA: TDOA, or Time Difference of Arrival, is a technique used in various applications, such as positioning and navigation systems, to determine the location of a signal source by measuring the time difference between the arrivals of the signal at different receivers. It relies on the principle that the time difference between the arrival of a signal at multiple receivers can be used to calculate the position of the signal source.
Transverse Axis: The transverse axis is a line that passes through the center of a conic section, such as an ellipse or hyperbola, and is perpendicular to the major axis. It represents the shorter of the two principal axes of the conic section and is crucial in understanding the properties and equations of these geometric shapes.
Vertical Transverse Axis: The vertical transverse axis of a hyperbola is the line that passes through the center of the hyperbola and is perpendicular to the transverse axis. It represents the shortest distance between the two branches of the hyperbola and is used to define the shape and orientation of the curve.
Vertices: Vertices are the points where two or more line segments or curves intersect to form the corners or endpoints of a geometric shape. They are the fundamental building blocks that define the structure and shape of various conic sections, including ellipses, hyperbolas, and rotated coordinate systems.
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