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
Top images from around the web for Graphing ellipses
Represented by the equation a2x2+[b](https://www.fiveableKeyTerm:b)2y2=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
when a>b ( along the x-axis)
when a<b (major axis along the y-axis)
Ellipses centered at a point ([h](https://www.fiveableKeyTerm:h),[k](https://www.fiveableKeyTerm:k)) other than the origin
Represented by the standard form equation a2(x−h)2+b2(y−k)2=1
Center of the 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 a2(x−h)2+b2(y−k)2=1 to obtain the specific equation for the ellipse
Given the vertices and of an ellipse
Calculate the center (h,k) by finding the midpoint between the vertices
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
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) 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
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=ac
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
: The distance from a focus to any point on the ellipse
Conic Sections and Related Concepts
: 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