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# X2 T03 01 Ellipse (2010)

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### X2 T03 01 Ellipse (2010)

1. 1. Conics
2. 2. Conics The locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix)
3. 3. Conics The locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix)
4. 4. Conics The locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix) e=0 circle
5. 5. Conics The locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix) e=0 circle e<1 ellipse
6. 6. Conics The locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix) e=0 circle e<1 ellipse e=1 parabola
7. 7. Conics The locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix) e=0 circle e<1 ellipse e=1 parabola e>1 hyperbola
8. 8. Ellipse (e < 1) y b A’ A -a a x -b
9. 9. Ellipse (e < 1) y b A’ A -a S a Z x -b
10. 10. Ellipse (e < 1) y b A’ A -a S a Z x -b SA = eAZ and SA’ = eA’Z
11. 11. Ellipse (e < 1) y b A’ A -a S a Z x -b SA = eAZ and SA’ = eA’Z (1) SA’ + SA = 2a (2) SA’ – SA = e(A’Z – AZ)
12. 12. Ellipse (e < 1) y b A’ A -a S a Z x -b SA = eAZ and SA’ = eA’Z (1) SA’ + SA = 2a (2) SA’ – SA = e(A’Z – AZ) = e(AA’) = e(2a) = 2ae
13. 13. b A’ A -a S a Z x -b (1) + (2); 2SA’ = 2a(1 + e) SA’ = a(1 + e)
14. 14. b A’ A -a S a Z x -b (1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e) SA’ = a(1 + e) SA = a(1 - e)
15. 15. b A’ A -a S a Z x -b (1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e) SA’ = a(1 + e) SA = a(1 - e) Focus OS = OA - SA
16. 16. b A’ A -a S a Z x -b (1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e) SA’ = a(1 + e) SA = a(1 - e) Focus OS = OA - SA = a – a(1 – e) = ae  S  ae,0 
17. 17. b A’ A -a S a Z x -b (1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e) SA’ = a(1 + e) SA = a(1 - e) Focus Directrix OS = OA - SA OZ = OA + AZ = a – a(1 – e) = ae  S  ae,0 
18. 18. b A’ A -a S a Z x -b (1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e) SA’ = a(1 + e) SA = a(1 - e) Focus Directrix OS = OA - SA OZ = OA + AZ SA = a – a(1 – e)  OA   SA  eAZ  = ae e  S  ae,0 
19. 19. b A’ A -a S a Z x -b (1) + (2); 2SA’ = 2a(1 + e) (1) - (2); 2SA = 2a(1 - e) SA’ = a(1 + e) SA = a(1 - e) Focus Directrix OS = OA - SA OZ = OA + AZ SA = a – a(1 – e)  OA   SA  eAZ  = ae e ae a1  e   S  ae,0    e e a a  directrices x    e e
20. 20. S ae,0  b P P  x, y  N A’ A -a a N , y a S Z x   e  -b
21. 21. S ae,0  b P P  x, y  N A’ A -a a N , y a S Z x   e  -b SP  ePN
22. 22. S ae,0  b P P  x, y  N A’ A -a a N , y a S Z x   e  -b SP  ePN 2  x  ae 2   y  02  e  x     y  y 2 a    e 2 2 a  x  ae   y  e  x   2 2  e
23. 23. S ae,0  b P P  x, y  N A’ A -a a N , y a S Z x   e  -b SP  ePN 2  x  ae 2   y  02  e  x     y  y 2 a    e 2 2 a  x  ae   y  e  x   2 2  e x 2  2aex  a 2 e 2  y 2  e 2 x 2  2aex  a 2 x 2 1  e 2   y 2  a 2 1  e 2 
24. 24. S ae,0  b P P  x, y  N A’ A -a a N , y a S Z x   e  -b SP  ePN 2  x  ae 2   y  02  e  x     y  y 2 a    e 2 2 a  x  ae   y  e  x   2 2  e x 2  2aex  a 2 e 2  y 2  e 2 x 2  2aex  a 2 x 2 1  e 2   y 2  a 2 1  e 2  x2 y2  2 1 a a 1  e  2 2
25. 25. b2 when x  0, y  b 1 a 1  e  i.e. 2 2 b 2  a 2 1  e 2 
26. 26. b2 when x  0, y  b 1 a 1  e  i.e. 2 2 b 2  a 2 1  e 2  Ellipse: (a > b) x2 y2 2  2 1 a b where; b 2  a 2 1  e 2  focus :  ae,0  a directrices : x   e e is the eccentricity major semi-axis = a units minor semi-axis = b units
27. 27. b2 when x  0, y  b 1 a 1  e  i.e. 2 2 b 2  a 2 1  e 2  Ellipse: (a > b) x2 y2 Note: If b > a 2  2 1 a b foci on the y axis where; b  a 1  e 2 2 2  a 2  b 2 1  e 2  focus :  ae,0  focus : 0,be  a b directrices : x   directrices : y   e e e is the eccentricity major semi-axis = a units minor semi-axis = b units
28. 28. b2 when x  0, y  b 1 a 1  e  i.e. 2 2 b 2  a 2 1  e 2  Ellipse: (a > b) x2 y2 Note: If b > a 2  2 1 a b foci on the y axis where; b  a 1  e 2 2 2  a 2  b 2 1  e 2  focus :  ae,0  focus : 0,be  a b directrices : x   directrices : y   e e e is the eccentricity major semi-axis = a units Area  ab minor semi-axis = b units
29. 29. e.g. Find the eccentricity, foci and directrices of the ellipse x2 y2   1 and sketch the ellipse showing all of the important 9 5 features.
30. 30. e.g. Find the eccentricity, foci and directrices of the ellipse x2 y2   1 and sketch the ellipse showing all of the important 9 5 features. x2 y2  1 9 5 a2  9 a3
31. 31. e.g. Find the eccentricity, foci and directrices of the ellipse x2 y2   1 and sketch the ellipse showing all of the important 9 5 features. x2 y2  1 b2  5 9 5 a 2 1  e 2   5 a2  9 a3
32. 32. e.g. Find the eccentricity, foci and directrices of the ellipse x2 y2   1 and sketch the ellipse showing all of the important 9 5 features. x2 y2  1 b2  5 9 5 a 2 1  e 2   5 91  e 2   5 a2  9 a3 5 1 e 2 9 4 e  2 9 2 e 3
33. 33. e.g. Find the eccentricity, foci and directrices of the ellipse x2 y2   1 and sketch the ellipse showing all of the important 9 5 features. x2 y2  1 b2  5 9 5 a 2 1  e 2   5 91  e 2   5 a2  9 2 a3  eccentricity  5 3 1 e 2 9 foci :  2,0  4 e  2 3 9 directrices : x  3  2 2 e 9 3 x 2
34. 34. y Auxiliary circle -3 3 x
35. 35. b 5 y a  3    Auxiliary circle -3 3 x
36. 36. b 5 y a  3    Auxiliary circle 5 -3 3 x  5
37. 37. b 5 y a  3    Auxiliary circle 5 -3 3 x  5
38. 38. b 5 y a  3    Auxiliary circle 5 -3 3 x  5
39. 39. b 5 y a  3    Auxiliary circle 5 -3 3 x  5
40. 40. b 5 y a  3    Auxiliary circle 5 -3 3 x  5
41. 41. b 5 y a  3    Auxiliary circle 5 -3 3 x  5
42. 42. b 5 y a  3    Auxiliary circle 5 -3 3 x  5
43. 43. b 5 y a  3    Auxiliary circle 5 -3 S’(-2,0) S(2,0) 3 x  5
44. 44. b 5 y a  3    Auxiliary circle 5 -3 S’(-2,0) S(2,0) 3 x  5 9 9 x x 2 2
45. 45. b 5 y a  3    Auxiliary circle 5 -3 S’(-2,0) S(2,0) 3 x  5 9 9 x x 2 2 Major axis = 6 units Minor axis  2 5 units
46. 46. (ii) 9 x 2  4 y 2  18 x  16 y  11  0
47. 47. (ii) 9 x 2  4 y 2  18 x  16 y  11  0 x 2  2 x y 2  4 y 11   4 9 36
48. 48. (ii) 9 x 2  4 y 2  18 x  16 y  11  0 x 2  2 x y 2  4 y 11   4 9 36  x  12  y  22 11 1 4     4 9 36 4 9
49. 49. (ii) 9 x 2  4 y 2  18 x  16 y  11  0 x 2  2 x y 2  4 y 11   4 9 36  x  12  y  22 11 1 4     4 9 36 4 9  x  12  y  22  1 4 9
50. 50. (ii) 9 x 2  4 y 2  18 x  16 y  11  0 x 2  2 x y 2  4 y 11   4 9 36  x  12  y  22 11 1 4     4 9 36 4 9  x  12  y  22  1 4 9 centre : (1,2)
51. 51. (ii) 9 x 2  4 y 2  18 x  16 y  11  0 x 2  2 x y 2  4 y 11   4 9 36  x  12  y  22 11 1 4     4 9 36 4 9  x  12  y  22  1 4 9 centre : (1,2) b2  9 b3
52. 52. (ii) 9 x 2  4 y 2  18 x  16 y  11  0 x 2  2 x y 2  4 y 11   4 9 36  x  12  y  22 11 1 4     4 9 36 4 9  x  12  y  22  1 4 9 centre : (1,2) b2  9 a 2  b 2 1  e 2  b3 4  91  e 2  5 e 3
53. 53. (ii) 9 x 2  4 y 2  18 x  16 y  11  0 x 2  2 x y 2  4 y 11   4 9 36  x  12  y  22 11 1 4     4 9 36 4 9  x  12  y  22  1 4 9 centre : (1,2) b2  9 a 2  b 2 1  e 2  b3 4  91  e 2  5 e 3 foci :  1,2  5  9 directrices : y  2  5
54. 54. (ii) 9 x 2  4 y 2  18 x  16 y  11  0 x 2  2 x y 2  4 y 11   4 9 36  x  12  y  22 11 1 4     4 9 36 4 9  x  12  y  22 Exercise 6A; 1, 2, 3, 5, 7,  1 4 9 8, 9, 11, 13, 15 centre : (1,2) b2  9 a 2  b 2 1  e 2  b3 4  91  e 2  5 e 3 foci :  1,2  5  9 directrices : y  2  5