1. "The set of all points such that the sum of the distance from two points is a constant. " (Mrs. Kirch/SSS Packet)
2. The equation for this conic section, the ellipse, is (x-h)²/a²+(y-k)²/b²=1 or (x-h)²/b²+(y-k)²/a²=1. The center will always be the h and k. H will be paired with the x and the k will be paired with the y. The a will always be bigger than the b. Where the a is placed will determine whether the graph is skinny or fat. If the a² is under the (x-h)², it will be fat/wide. If the a² is under the (y-k)², it will be skinny/long. The equation will always be equal to 1 and the equation will always be added.The ellipse on a graph looks like an elongated circle. It can look fat/wide or it can look skinny/tall.
The key features on the ellipse are the center, two vertices, two co-vertices, two foci, major axis, minor axis, and the position/look of the ellipse. The key features in the equation that will be needed to find the key features on the graph are the standard form, center, a, b, c, and the eccentricity. The easiest way to graph an ellipse is to obtain the standard form first. To find the equation, you complete the square. After doing so, divide everything by the number that is on the right side so the equation will be equal to one. After that, identify the circle by looking at both numerators. The h will be associated with the x and the k will be associated with the y. Since the numerators are (x-h)² and (y-k)². That means the center is just (h, k). However, if your standard form has a (x+h)², then the h will be a negative; the same concept applies to the y and the center may be (-h, -k) depending on which numerator has addition. Then plot the center. Now go back to the equation and determine if the graph is going to be fat/wide or skinny/tall. The graph will be fat/wide if the a² is the denominator for the (x-h)². The graph will be skinny if the a² is the denominator for (y-k)². After identifying the shape of the graph, find your a and b. Since in the equation is is squared, you need to square root it to find the distance from the center to one endpoint of the major/minor axis and the other endpoint of the major/minor axis.Now that a and b have been found, to find the vertices and co-vertices (a is associated with vertices and b is a associated with co-vertices) we need to observe the shape of the graph again. If it is wide /fat, the y (meaning the y from the center) in the ordered pair for the vertices will stay the same and the x (meaning the x from the center) in the ordered pair so you have to add and subtract the y from the center to get the other number for the ordered pair. If it is wide/fat, the x in the co-vertices will remain the same so we have to add and subtract the y from the center to get the other numbers for the ordered pair.The same thing goes for skinny/tall but vice-versa meaning the a in the vertices stays the same and the y in the co-vertices stays the same. Plot the vertices and connect the two endpoint with a solid line and plot the co-vertices and connect the two endpoint with a dashed line, that way it will be easier to identify which line is which. Now just draw you ellipse and you are done! Just kidding! But you're almost there! Now we have to find c, which will help us in finding the foci. To find c, we use the equation a²-b²=c². After plugging in the equation and finding the c, we need to find the ordered pair of it so we can graph it. Look at the vertices and whatever number stays, the x or the y, then the same will go for the foci since the foci will be on the major axis like the vertices. Now just add or subtract from the other number, x or y depending on the shape, and plot. The purpose of the foci is to know the how fat/wide or how skinny/tall it is since the sum of two distance from the foci is a constant. The farther from the center, the more elongated it will be and if closer to the center it will be less elongated. Now we need one more thing! The eccentricity! to find the eccentricity, we need to use the equation c/a. The number should be between 0 to 1. Since it is in between 0 and 1, then that means an ellipse deviates from being a circle a little bit.
3. Conics of ellipses can be seen where very few men have gone: space. The rotation of the planets around the sun form an ellipse. Another space example would be Haley's Comet. It has a rotation that is shaped like an ellipse. An ellipse that isn't so out of this world is a glass of water. When tilted slightly, the surface of the water can be viewed to look like an ellipse. Ellipses can be found in common places as well: buildings, statues, car logos, and even more places! Can you find all of them?
4. Works Cited:
http://britton.disted.camosun.bc.ca/jbconics.htm
http://www.youtube.com/watch?v=7mqUYWtRL5Y
http://intmstat.com/plane-analytic-geometry/earth.jpg
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