RWA #1: Unit M Concept 5a: Ellipses

1. Definition: "The set of all points such that the sum of the distance from two points is a constant."

http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/ellipse-drawn-from-definition-geogebra-dynamic-worksheet

2. Description:

Algebraically- There are two equations which ellipses can be written as which are
                        (x-h)^2/a^2 + (y-k)^2/ b^2 =1 or (x-h)^2/b^2 + (y-k)^2 / a^2 =1

Graphically-  On a graph, an ellipse looks like an outstretched circle either looking skinny or fat. There 

                      is a major axis and a minor axis. On the major axis, there are two points called the foci 

                      and two vertices. On the minor axis, there are two points known as the co-vertices. 

Key Parts: 

Above there is a sketch of an ellipse. We can see that it is a "fat" ellipse  so therefore we know "a^2" is going to be the first denominator. We know what the center of an ellipse is by using the (h,k) rule on the standard equation. Based on the denominators, we know what "a" is depending if its value is larger than the other denominator. The larger value is "a", therefore making the other value "b." To find these on the graph, it basically all depends whether it is horizontal or vertical. A and B will be the distance away from the center, reaching to the vertex or co-vertex. To ind "c" we must use the equation a^2 - b^2 = c^2. From there, you are able to find the eccentricity by using c/a. 

Foci/eccentricity: 

To find c, we know that we must use the equation a^2 - b^2 = c^2. Now that yo have the value of c, you divide that over a which gives you your eccentricity. Your eccentricity should be greater than 0 but less than 1. The foci will always be on the major axis and be inside the ellipse, not outside. 

3. Real World Application 

The most well known example of an ellipse can be the orbit of satellites such as the earth. We are able to see how the rotation of the earth around the sun is in an elliptical pathway. The rotation of the sun creates the four different seasons we experience. In the video it says, "In the spring equinox, the overhead sun is over the equator which receives the most amount of solar radiation." http://www.youtube.com/watch?v=WLRA87TKXLM  In an article it says, " Earth's orbit is almost a circle, it has an eccentricity less than 0.02."  http://www.universetoday.com/61202/earths-orbit-around-the-sun/ With this we know that the orbit of the earth is not a perfect circle because the eccentricity is more than 0 making it an ellipse.

4. Works Cited: 

http://www.universetoday.com/61202/earths-orbit-around-the-sun/

http://www.youtube.com/watch?v=WLRA87TKXLM

http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/ellipse-drawn-from-definition-geogebra-dynamic-worksheet