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# 1003 ch 10 day 3

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• ### 1003 ch 10 day 3

1. 1. 10.2 The EllipsePsalm 63:5 "My soul will be satisﬁed as with fat and richfood, and my mouth will praise you with joyful lips,"
2. 2. Another way to deﬁne the ellipse is this:An Ellipse is the set ofpoints in the plane thesum of whose distancesfrom two ﬁxed points(the Foci), is a constant. use K on keyboard to Pause/Play
3. 3. Another way to deﬁne the ellipse is this:An Ellipse is the set ofpoints in the plane thesum of whose distancesfrom two ﬁxed points(the Foci), is a constant. How narrow or wide the ellipse is will be called its eccentricity. More on this later. use K on keyboard to Pause/Play
4. 4. Another way to deﬁne the ellipse is this:An Ellipse is the set ofpoints in the plane thesum of whose distancesfrom two ﬁxed points(the Foci), is a constant. How narrow or wide the ellipse is will be called its eccentricity. More on this later. use K on keyboard to Pause/Play
5. 5. What follows is the derivation of the equation for an ellipse centered at the origin.
6. 6. What follows is the derivation of the equation for an ellipse centered at the origin.You don’t need to write any of this down, but I want you to pay careful attention and understand the work.
7. 7. Consider this ellipse centered at the Origin with foci onthe x-axis:and let the sum of F1P and F2 P be the constant 2awhich means ± a are the x-intercepts and a > c
8. 8. Consider this ellipse centered at the Origin with foci onthe x-axis:and let the sum of F1P and F2 P be the constant 2awhich means ± a are the x-intercepts and a > c then 2 2 2 2 ( x + c) +y + ( x − c) + y = 2a
9. 9. from the previous slide 2 2 2 2( x + c) +y + ( x − c) + y = 2a
10. 10. from the previous slide 2 2 2 2( x + c) +y + ( x − c) + y = 2a 2 2 2 2( x − c) + y = 2a − ( x + c) +y
11. 11. from the previous slide 2 2 2 2( x + c) +y + ( x − c) + y = 2a 2 2 2 2( x − c) + y = 2a − ( x + c) +ysquare both sides and expand
12. 12. from the previous slide 2 2 2 2 ( x + c) +y + ( x − c) + y = 2a 2 2 2 2 ( x − c) + y = 2a − ( x + c) +y square both sides and expand 2 2 2 2 2 2 2 2 2x − 2cx + c + y = 4a − 4a ( x + c) + y + x + 2cx + c + y
13. 13. from the previous slide 2 2 2 2 ( x + c) +y + ( x − c) + y = 2a 2 2 2 2 ( x − c) + y = 2a − ( x + c) +y square both sides and expand 2 2 2 2 2 2 2 2 2x − 2cx + c + y = 4a − 4a ( x + c) + y + x + 2cx + c + y
14. 14. from the previous slide 2 2 2 2 ( x + c) +y + ( x − c) + y = 2a 2 2 2 2 ( x − c) + y = 2a − ( x + c) +y square both sides and expand 2 2 2 2 2 2 2 2 2x − 2cx + c + y = 4a − 4a ( x + c) + y + x + 2cx + c + y 2 2 2 4a ( x + c) + y = 4a + 4cx
15. 15. from the previous slide 2 2 2 2 ( x + c) +y + ( x − c) + y = 2a 2 2 2 2 ( x − c) + y = 2a − ( x + c) +y square both sides and expand 2 2 2 2 2 2 2 2 2x − 2cx + c + y = 4a − 4a ( x + c) + y + x + 2cx + c + y 2 2 2 4a ( x + c) + y = 4a + 4cx 2 2 2 a ( x + c) + y = a + cx
16. 16. from the previous slide 2 2 2a ( x + c) + y = a + cx
17. 17. from the previous slide 2 2 2a ( x + c) + y = a + cx square both sides
18. 18. from the previous slide 2 2 2 a ( x + c) + y = a + cx square both sidesa 2 (( x + c ) + y ) = a 2 2 4 2 2 + 2a cx + c x 2
19. 19. from the previous slide 2 2 2 a ( x + c) + y = a + cx square both sides a 2 (( x + c ) + y ) = a 2 2 4 2 + 2a cx + c x 2 2 2 2 2 2 2 2 2 4 2 2 2a x + 2a cx + a c + a y = a + 2a cx + c x
20. 20. from the previous slide 2 2 2 a ( x + c) + y = a + cx square both sides a 2 (( x + c ) + y ) = a 2 2 4 2 + 2a cx + c x 2 2 2 2 2 2 2 2 2 4 2 2 2a x + 2a cx + a c + a y = a + 2a cx + c x
21. 21. from the previous slide 2 2 2 a ( x + c) + y = a + cx square both sides a 2 (( x + c ) + y ) = a 2 2 4 2 + 2a cx + c x 2 2 2 2 2 2 2 2 2 4 2 2 2a x + 2a cx + a c + a y = a + 2a cx + c x 2 2 2 2 2 2 4 2 2 a x +a c +a y =a +c x
22. 22. from the previous slide 2 2 2 a ( x + c) + y = a + cx square both sides a 2 (( x + c ) + y ) = a 2 2 4 2 + 2a cx + c x 2 2 2 2 2 2 2 2 2 4 2 2 2 a x + 2a cx + a c + a y = a + 2a cx + c x 2 2 2 2 2 2 4 2 2 a x +a c +a y =a +c xsubtract each of these terms from both sides and group
23. 23. from the previous slide 2 2 2 a ( x + c) + y = a + cx square both sides a 2 (( x + c ) + y ) = a 2 2 4 2 + 2a cx + c x 2 2 2 2 2 2 2 2 2 4 2 2 2 a x + 2a cx + a c + a y = a + 2a cx + c x 2 2 2 2 2 2 4 2 2 a x +a c +a y =a +c xsubtract each of these terms from both sides and group (a x 2 2 − c x )+ a y = a − a c 2 2 2 2 4 2 2
24. 24. from the previous slide 2 2 2 a ( x + c) + y = a + cx square both sides a 2 (( x + c ) + y ) = a 2 2 4 2 + 2a cx + c x 2 2 2 2 2 2 2 2 2 4 2 2 2 a x + 2a cx + a c + a y = a + 2a cx + c x 2 2 2 2 2 2 4 2 2 a x +a c +a y =a +c xsubtract each of these terms from both sides and group (a x 2 2 − c x )+ a y = a − a c 2 2 2 2 4 2 2 x (a 2 2 − c ) + a y = a (a − c 2 2 2 2 2 2 )
25. 25. from the previous slidex (a − c ) + a y = a (a − c 2 2 2 2 2 2 2 2 )
26. 26. from the previous slide x (a − c ) + a y = a (a − c 2 2 2 2 2 2 2 2 )recall that a > c , so a − c > 0 and 2 2 2 (we can divide by a a − c2 2 )
27. 27. from the previous slide x (a − c ) + a y = a (a − c 2 2 2 2 2 2 2 2 )recall that a > c , so a − c > 0 and 2 2 2 (we can divide by a a − c2 2 ) 2 2 x y 2 + 2 2 =1 a a −c
28. 28. from the previous slide x (a − c ) + a y = a (a − c 2 2 2 2 2 2 2 2 )recall that a > c , so a − c > 0 and 2 2we can divide by a a − c 2 2 (2 ) 2 2 x y 2 + 2 2 =1 a a −cthen we let b = a − c 2 2 2 ( and b < a )to get the familiar:
29. 29. from the previous slide x (a − c ) + a y = a (a − c 2 2 2 2 2 2 2 2 )recall that a > c , so a − c > 0 and 2 2we can divide by a a − c 2 2 2 ( ) 2 2 x y 2 + 2 2 =1 a a −cthen we let b = a − c 2 2 2 ( and b < a )to get the familiar: 2 2 x y 2 + 2 =1 a b
30. 30. 2 2 x y 2 + 2 =1 a bmajor axis has length 2aminor axis has length 2b
31. 31. 2 2 x y 2 + 2 =1 b amajor axis has length 2aminor axis has length 2b
32. 32. To sketch the graph of an ellipse, ﬁnd the lengths ofthe major and minor axes ... and sketch away! Pleaseinclude the foci.
33. 33. Example: Sketch the graph of 4x 2 + 25y 2 = 100
34. 34. Example: Sketch the graph of 4x 2 + 25y 2 = 100 divide by the constant to put it into standard form
35. 35. Example: Sketch the graph of 4x 2 + 25y 2 = 100 divide by the constant to put it into standard form 2 2 x y + =1 25 4
36. 36. Example: Sketch the graph of 4x 2 + 25y 2 = 100 divide by the constant to put it into standard form 2 2 x y + =1 25 4 2 a = 25 a = ±5
37. 37. Example: Sketch the graph of 4x 2 + 25y 2 = 100 divide by the constant to put it into standard form 2 2 x y + =1 25 4 2 2 a = 25 b =4 a = ±5 b = ±2
38. 38. Example: Sketch the graph of 4x 2 + 25y 2 = 100 divide by the constant to put it into standard form 2 2 x y + =1 25 4 2 2 2 2 2 a = 25 b =4 c = a −b 2 a = ±5 b = ±2 c = 25 − 4 c = ± 21
39. 39. Example: Sketch the graph of 4x 2 + 25y 2 = 100 divide by the constant to put it into standard form 2 2 x y + =1 25 4 2 2 2 2 2 a = 25 b =4 c = a −b 2 a = ±5 b = ±2 c = 25 − 4 c = ± 21
40. 40. HW #3“As a rule of thumb, involve everyone in everything.” Tom Peters