Glossary

11.3 Ellipses

4 min read•june 25, 2024

Ellipses are fascinating curves with unique properties. They're defined by equations involving x and y coordinates, and their shape depends on the lengths of their axes. Ellipses can be centered at the origin or any other point on coordinate plane.

These curves have real-world applications in planetary orbits, architecture, and technology. Key components of ellipses include the , , , and axes. Understanding these elements helps us analyze and apply ellipses in various fields.

Ellipse Fundamentals

Graphing ellipses

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  • Ellipses centered at the origin (0,0)(0, 0)
    • Represented by the equation x2a2+y2[b](https://www.fiveableKeyTerm:b)2=1\frac{x^2}{a^2} + \frac{y^2}{[b](https://www.fiveableKeyTerm:b)^2} = 1, where aa and bb are the lengths of the semi-major and semi-minor axes respectively
    • Orientation determined by the relative values of aa and bb
      • when a>ba > b ( along the x-axis)
      • when a<ba < b (major axis along the y-axis)
  • Ellipses centered at a point ([h](https://www.fiveableKeyTerm:h),[k](https://www.fiveableKeyTerm:k))([h](https://www.fiveableKeyTerm:h), [k](https://www.fiveableKeyTerm:k)) other than the origin
    • Represented by the standard form equation (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
    • Center of the translated to the point (h,k)(h, k)
    • Maintains the same shape and orientation as an ellipse centered at the origin with the same aa and bb values

Equations of ellipses

  • Given the center (h,k)(h, k), and the lengths of the semi-major and semi-minor axes (a,b)(a, b)
    • Plug in the values into the standard form equation (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 to obtain the specific equation for the ellipse
  • Given the vertices and of an ellipse
    1. Calculate the center (h,k)(h, k) by finding the midpoint between the vertices
    2. Calculate the lengths of the semi-major and semi-minor axes by measuring the distance from the center to the vertices and co-vertices respectively
    3. Substitute the obtained values into the standard form equation to derive the equation of the ellipse

Ellipse Applications and Components

Real-world applications of ellipses

  • Planetary orbits
    • Planets orbit the sun in elliptical paths with the sun located at one of the foci (Kepler's first law of planetary motion)
  • Whispering galleries
    • Elliptical rooms (U.S. Capitol Building, St. Paul's Cathedral) designed so that a whisper at one focus can be heard clearly at the other focus due to the reflective properties of the elliptical shape
  • Reflective properties
    • Ellipses have the unique property that light or sound waves emanating from one focus will reflect off the ellipse's boundary and converge at the other focus (used in , satellite dish design)

Components of ellipses

  • Center: The point (h,k)(h, k) at the center of the ellipse
  • Vertices: The points on the ellipse farthest from the center, located at the ends of the major axis
  • Co-vertices: The points on the ellipse closest to the center, located at the ends of the
  • Foci: Two points inside the ellipse with the property that the sum of the distances from any point on the ellipse to the foci is constant
    • Foci are located on the major axis, equidistant from the center
    • Distance from the center to each focus calculated by the formula [c](https://www.fiveableKeyTerm:c)=a2−b2[c](https://www.fiveableKeyTerm:c) = \sqrt{a^2 - b^2}
  • Major axis: The longest diameter of the ellipse, passing through the center, vertices, and foci
  • Minor axis: The shortest diameter of the ellipse, passing through the center and co-vertices, perpendicular to the major axis
  • : A value between 0 and 1 that measures how much an ellipse deviates from a circle, calculated by the formula e=cae = \frac{c}{a}
    • e=0e = 0 represents a circle, while values approaching 1 represent increasingly elongated ellipses
    • Eccentricity of Earth's orbit around the sun is approximately 0.0167, making it nearly circular
  • : The distance from a focus to any point on the ellipse
  • : A family of curves obtained by intersecting a plane with a double cone, including circles, ellipses, parabolas, and hyperbolas
  • : A line used in the definition of conic sections, where the ratio of the distance from any point on the conic to a focus and the distance to the directrix is constant (eccentricity)
  • : A chord of the ellipse passing through a focus and perpendicular to the major axis
  • : Three fundamental laws describing the motion of planets around the sun, with the first law stating that planetary orbits are elliptical with the sun at one focus

Key Terms to Review (27)

A: The variable 'a' is a fundamental component in the study of quadratic equations and ellipses. It represents a constant value that is used in the mathematical expressions and formulas related to these topics.
B: The variable 'b' is a commonly used parameter that represents a constant value or coefficient in various mathematical contexts. It is a fundamental component in equations and formulas, playing a crucial role in the analysis and understanding of linear, quadratic, and elliptical relationships.
C: The variable 'c' is a commonly used letter to represent a constant or a coefficient in various mathematical contexts, including the solving of quadratic equations using the quadratic formula and the equation of an ellipse. It serves as a placeholder for a specific numerical value that remains fixed throughout the problem or equation.
Center: The center of a geometric shape, such as a circle, ellipse, or hyperbola, is the point that is equidistant from all points on the perimeter or boundary of the shape. It is the fixed point around which the shape is symmetrically arranged.
Co-vertices: Co-vertices, in the context of an ellipse, refer to the points on the ellipse where the major and minor axes intersect. These points represent the maximum and minimum values of the ellipse's coordinates, defining its overall shape and size.
Completing the Square: Completing the square is a technique used to solve quadratic equations by transforming them into a perfect square form. This method involves adding a constant to both sides of the equation to create a perfect square on one side, allowing for easier factorization and solution of the equation.
Conic Sections: Conic sections are the curves formed by the intersection of a plane with a cone. These curves include the circle, ellipse, parabola, and hyperbola, and they have numerous applications in mathematics, science, and engineering.
Directrix: The directrix is a fixed, straight line that, along with the focus, defines the shape and position of a conic section, such as a parabola, ellipse, or hyperbola. It serves as a reference point for the curve and is used in the mathematical equations that describe these geometric shapes.
Eccentricity: Eccentricity is a measure of the deviation of a conic section, such as an ellipse or hyperbola, from a perfect circle. It quantifies how much the shape of the curve deviates from a circular shape, providing important information about the shape and properties of these geometric figures.
Ellipse: An ellipse is a closed, two-dimensional geometric shape that resembles an elongated circle. It is defined as the set of all points on a plane where the sum of the distances from two fixed points, called the foci, is constant.
Focal Radius: The focal radius is a key concept in the study of ellipses, which are closed, two-dimensional shapes with two focal points. The focal radius refers to the distance between the center of the ellipse and one of its focal points, and it is a fundamental parameter that determines the shape and properties of the ellipse.
Foci: The foci (plural of focus) are the fixed points within an ellipse or hyperbola that define the shape and orientation of these conic sections. They are essential in understanding the properties and equations of these geometric figures.
General Form: The general form is a standardized way of expressing mathematical equations or expressions that allows for easy identification and manipulation of their underlying structure. This term is particularly relevant in the contexts of solving quadratic equations by completing the square and understanding the properties of ellipses.
H: The term 'h' refers to the distance between the center of an ellipse and the point where the ellipse intersects the x-axis. This distance is a crucial parameter that, along with the term 'a', defines the size and shape of an ellipse.
Horizontal Ellipse: A horizontal ellipse is a type of conic section that is defined as the set of all points in a plane where the sum of the distances from two fixed points, called the foci, is constant. Unlike a circle, which is a special case of an ellipse where the two foci coincide, an ellipse has two distinct foci that are not equidistant from the center.
K: The variable 'k' is a commonly used letter in mathematics to represent an unknown or a constant value. In the context of the topic 11.3 Ellipses, 'k' is often used to represent specific parameters or characteristics of an ellipse.
Kepler's Laws of Planetary Motion: Kepler's laws of planetary motion are three fundamental principles that describe the motion of planets around the Sun. These laws, formulated by the German astronomer Johannes Kepler in the early 17th century, provide a mathematical framework for understanding the dynamics of the solar system, particularly in the context of ellipses.
Latus Rectum: The latus rectum, or parameter, of a conic section is a line segment that passes through the focus of the conic and is perpendicular to the major axis. It provides important information about the size and shape of the conic section, and its value is a key characteristic in the equations describing parabolas, ellipses, and hyperbolas.
Lithotripsy: Lithotripsy is a non-invasive medical procedure used to treat kidney stones and other types of stones in the urinary tract. It involves the use of focused shock waves to break up these stones into smaller fragments that can be more easily passed through the body.
Major Axis: The major axis of an ellipse is the longest diameter of the ellipse, passing through the center and extending to the points where the ellipse intersects the x-axis. It represents the maximum width or length of the elliptical shape.
Minor Axis: The minor axis of an ellipse is the shorter of the two perpendicular lines that intersect at the center of the ellipse, dividing it into two equal halves. It is one of the defining characteristics of an elliptical shape.
Semi-Major Axis: The semi-major axis is one of the defining parameters of an ellipse, which is a closed, two-dimensional shape that resembles an elongated circle. It represents the length of the longer of the two principal axes of the ellipse, and is a crucial component in understanding the size and shape of an elliptical object or orbit.
Semi-Minor Axis: The semi-minor axis is one of the two principal axes of an ellipse, the other being the semi-major axis. It represents the shorter of the two axes and is essential in defining the shape and size of an ellipse.
Standard Form: The standard form of an equation is a specific way of writing the equation that provides a clear and organized structure, making it easier to analyze and work with the equation. This term is particularly relevant in the context of linear equations, quadratic equations, and other polynomial functions.
Vertical Ellipse: A vertical ellipse is a type of ellipse where the major axis is oriented vertically, perpendicular to the horizontal axis. This geometric shape is defined by two unequal radii, with the longer radius representing the major axis and the shorter radius representing the minor axis.
Vertices: The vertices of a geometric shape refer to the points where two or more line segments, curves, or edges meet. They are the corners or points that define the shape and its boundaries.
Whispering Gallery: A whispering gallery is an architectural feature that allows sound to be transmitted with great clarity over a large distance through curved surfaces. This acoustic phenomenon occurs in enclosed spaces where sound waves travel along the walls and can be heard distinctly at a distant point, even when spoken at a low volume.
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