Geometric definition of an ellipse - The set of all points in a plane whose sum of distances from 2 fixed points called foci (plural of focus) is constant. The shape of an ellipse is like a circle that got squashed so it resembles a football. Inside the ellipse are two fixed points called foci. Pick any point on the ellipse and find the distance from it to focus 1, call it d1. From that same point, find the distance from it to focus 2, call it d2. The sum of d1 + d2 will always add to the same number no matter what point you chose to pick. Just like with the parabolas, these problems rely on knowing the general format of the equations and applying them based on the given information. x^2/a^2 + y^2/b^2 = 1. Equation of an ellipse centered at the origin. The general formats for the equations of ellipses centered at the origin and with shifts are in the word document below. The orientation of the ellipse, whether elongated more along the vertical axis or the horizontal axis is determined by which is the bigger denominator, a^2 or b^2. If the bigger denominator is under the x, then the main axis (longest) is the horizontal. If the biggest denominator is under the y, then the main axis (longest) is the vertical. So when graphing, the important thing is which coordinate is over the larger denominator, x or y. Below are some links showing the basic graph of an ellipse, and to solve some problems. Also included are notes with problems worked out and the answer key with all the homework problems worked out in detail. As always, please email me with any specific questions. Some time by the end of this week or early next week, you will have a short assessment on these problems. Homework due Friday: pages 759-60: 2, 4, 6, 10, 14, 20, 30 pages 781-82: 2, 4, 16 ( Hint for when you have difficulty getting the coefficient in front of the x^2 or y^2 to 1 when the right side is already 1 e:g: 9x^2 / 1 + 10y^2 / 1 = 1) Change the coefficient in front of the x^2 or y^2 to 1 and represent the denominator as 1/Coefficient of what had been in front of the x or y. The above equation is equal to: x^2 / (1/9) + y^2 / 1/10 = 1 This is now in standard form. Link for ellipses. Basic concepts. Parts 1 and 2 www.youtube.com/watch?v=LVumLCx3fQo www.youtube.com/watch?v=oZB69DY0q9A Graphing ellipses centered at the origin with major axis along x and along y www.youtube.com/watch?v=azI5kALyiXs&feature=youtu.be (horizontal axis) www.youtube.com/watch?v=3qckea8OuN8&feature=youtu.be (major axis vertical) Graphs not centered at the origin: www.youtube.com/watch?v=tJmp1PJD9o8&feature=youtu.be www.youtube.com/watch?v=oWGyVpq94CM&feature=youtu.be Given an equation, rewrite in standard form and graph. www.youtube.com/watch?v=-i48L0WQ-2I&feature=youtu.be www.youtube.com/watch?v=CL7SNu-5riw&feature=youtu.be Given some information about the ellipse (e.g focus, vertex, etc), find the equation. www.youtube.com/watch?v=k5gMLDjMfvI&feature=youtu.be www.youtube.com/watch?v=RWaEIJOlHlw&feature=youtu.be There are a number of videos on this website for ellipses: www.mathispower4u.com/alg-2.php in addition to the ones posted above.
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