11.3 Ellipses

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 a coordinate plane.

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

Ellipse Fundamentals

Graphing ellipses

• Ellipses centered at the origin $(0, 0)$
  • Represented by the standard form equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axes respectively
  • Orientation determined by the relative values of $a$ and $b$
    • Horizontal ellipse when $a > b$ (major axis along the x-axis)
    • Vertical ellipse when $a < b$ (major axis along the y-axis)
• Ellipses centered at a point $(h, k)$ other than the origin
  • Represented by the standard form equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
  • Center of the ellipse translated to the point $(h, k)$
  • Maintains the same shape and orientation as an ellipse centered at the origin with the same $a$ and $b$ values

Equations of ellipses

• Given the center $(h, k)$, and the lengths of the semi-major and semi-minor axes $(a, b)$
  • Plug in the values into the standard form equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ to obtain the specific equation for the ellipse
• Given the vertices and co-vertices of an ellipse
  1. Calculate the center $(h, k)$ 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 lithotripsy, satellite dish design)

Components of ellipses

• Center: The point $(h, k)$ 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 minor axis
• 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 $c = \sqrt{a^2 - b^2}$
• 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
• Eccentricity: A value between 0 and 1 that measures how much an ellipse deviates from a circle, calculated by the formula $e = \frac{c}{a}$
  • $e = 0$ 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
• Focal radius: The distance from a focus to any point on the ellipse

Conic Sections and Related Concepts

• Conic sections: A family of curves obtained by intersecting a plane with a double cone, including circles, ellipses, parabolas, and hyperbolas
• Directrix: 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)
• Latus rectum: A chord of the ellipse passing through a focus and perpendicular to the major axis
• Kepler's laws of planetary motion: 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