1. An ellipse is mathematically defined by two foci. The two foci can hit any point on the ellipse, and when the distance of both are added, it will equal a constant. In the picture, there are many dashed lines hitting a point on the ellipse, but also hitting both foci. When the distances of the two lines are added, all the points will be a constant.
2. The foci affects the eccentricity. The eccentricity of an ellipse is 0>x>1, but in many problems, the eccentricity is always .5 and greater. This is okay because it is still within the boundaries of 0 to 1, but shouldn't eccentricity be .1 or .2 or .3 or .4, too? I've come to the conclusion that an ellipse can be .4 or less, but then the "a" and "b" would be fairly close to one another (and when I mean fairly close, I mean .000000001 close). This would make the ellipse almost a circle, but not quite yet. If it were to be a small eccentricity, the foci would be a lot closer to the center rather than to the vertices.
As you can see in the picture, when the eccentricity gets smaller that ellipse becomes rounder and the foci get closer the the center.
3. An ellipse can be found in many places of the world. But, one of the most important things that take shape of an ellipse are the orbits of the planets. It is very important that it is in a shape of an ellipse because then, the planets will never collide with one another. Football stadiums also take the shape of an ellipse, fitting thousands upon thousands of people inside. Click here for real life ellipse. Enjoy!
Citations:
Ellipse with distances pasted from: http://www.wyzant.com/help/images/conic18.gif
Ellipse with different foci pasted from: http://hrsbstaff.ednet.ns.ca/joldford/Handouts/Kepler%20Lab_files/image002.gif
Website for real life applications: http://bit.ly/TnIa82
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