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# Ellipses

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### Ellipses

1. 1. Algebra 2 Warm up 3.10.14 Find the center and radius of the circle and then graph it: x  y  8 x  6 y  11 2 2
2. 2. And now it’s time for.. The Lame Joke of the day.. What did Sushi A say to Sushi B? Wasabi! (What’s up B, get it?)
3. 3. And now it’s time for.. The Lame Joke of the day.. What did one ocean say to the other ocean? Nothing , it just waved
4. 4. THE
5. 5. An ellipse is the set of all points such that the sum of the distances from two fixed points, called the foci, is a constant. You can draw an ellipse by taking two push pins in cardboard with a piece of string attached as shown: The place where each pin is is a focus (the plural of which is foci). The sum of the distances from the ellipse to these points stays the same because it is the length of the string.
6. 6. Minor Axis The Ellipse Focus 1 Point Major Axis Focus 2 PF1 + PF2 = constant 3.4.2
7. 7. PARTS OF AN ELLIPSE The major axis is in the direction of the longest part of the ellipse major axis foci minor axis center The vertices are at the ends of the major axis The foci are always on the major axis foci
8. 8. The standard equation for an ellipse centered at the origin: 2 2 x y 2 2 2  2  1, where a  b  0 and a  b  c 2 a b The values of a, b, and c tell us about the size of our ellipse. b a c b a c
9. 9. x2 y2 Find the vertices and foci and graph the ellipse:  1 9 4 From the center the The ends of this axis are the vertices ends of major axis are "a" each direction. "a" is the square root of this value a a b (-3, 0)  We can now draw the ellipse 5 ,0  b 5 ,0 (3, 0) To find the foci, they are "c" away from the center in each direction. Find "c" by the equation: c  9  4  5 c  5  2.2 2 From the center the ends of minor axis are "b" each direction. "b" is the square root of this value c2  a2  b2
10. 10. 2 2 22 2 In 2 x y yy22 An ellipse can have a vertical major axis. x xx  y  1  1 1   222  1 22 2 that case the a2 is under the y2 1 4a  1 bb 164 You can tell which value is a because a2 is always greater than b2 From the (0, 4) Find the center, the equation of vertices the ellipse are 4 each shown way so "a" is 4. (0, -4) From the center the ends of minor axis are 1 each direction so "b" is 1 To find the foci, they are "c" away from the center in each direction along the major axis. Find "c" by the equation: c2  a2  b2 2 c  16  1  15 c  15  3.9
11. 11. Graph the ellipse: x2 y2  1 16 64 Is the ellipse horizontal or vertical? Vertical because the larger number is under “y” a is always the larger number: a  64  8 b  16  4
12. 12. Graph the ellipse: x 2  25 y 2  25 This equation is not in standard form (equal to 1) , so we must divide both sides by 25. x2 2  y 1 25 Is the ellipse horizontal or vertical? Horizontal because the larger number is under “x” a is always the larger number. a  25  5 b  1 1
13. 13. Graph each ellipse: 9 x 2  y 2  36 x 2  25 y 2  25
14. 14. There are many applications of ellipses.
15. 15. A particularly interesting one is the whispering gallery. The ceiling is elliptical and a person stands at one focus of the ellipse and can whisper and be heard by another person standing at the other focus because all of the sound waves that reach the ceiling from one focus are reflected to the other focus.