An ellipse is a shape defined by two focal points and a taut string. Drawing an ellipse involves placing two points on a page and using a string and pencil to trace the shape while keeping the string taut. Ellipses have the property that light rays or sound waves bouncing off the curve will reflect back to the two focal points, explaining whispering galleries where people standing at the focal points can hear each other clearly. The document provides examples of writing the equation of ellipses given properties like the center, vertices, or axes.

1. 10.4 Ellipses

2. http://www.mathwarehouse.com/ellipse/eccentricity-orbiting-planets.php An Ellipse is:

4. The location of each tack is called a focus, because one of the properties possessed by an ellipse: light rays emanating from a source at one focus and reflected by the ellipse would all return to the other focus. This we prove by recalling a physical property of light: it always travels by the shortest available path.

7. This reflecting property of the ellipse explains the sound effects obtained in certain so-called whispering galleries. If you and your friend stand close to and facing the wall, at opposite corners of the intersection of the two ramps, as at A and B in Figure 21, you can talk in very subdued tones and hear each other perfectly, while people passing between you will be unable to hear your conversation. You are standing at the two foci of an elliptical part of the ceiling.

9. Cincinnati’s Union Terminal

14. Example 1. Write the equation of an ellipse with center of (0, 0), vertex at (0, 5), and co-vertex at (2, 0).

15. Example 2 . Write the equation of the ellipse with center (-2, 1), horizontal major axis of length 6 and minor axis of length 4.