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Hyperbola: Focal Distance Difference

Hyperbola: Focal Distance Difference
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2016-02-06 10:22
FunctionCoordinate GeometryTransformation
In collections Conic Sections Conic Sections- Maths Made Easy
This example demonstrates the geometric properties of a hyperbola, showing the foci, vertices, and focal distance difference. Using GeoGebra commands such as Focus, Vertex, Translate, etc., the construction process of the hyperbola can be dynamically observed. Suitable for high school mathematics education, helping students understand the geometric properties and related concepts of hyperbolas.

GeoGebra Commands & Parameters

  1. Focus(c) → F_1, F_2
  2. Vertex(c) → C, D, E, F
  3. Segment(C, D) → d
  4. Translate(C, (0, b + 4)) → C'
  5. Translate(D, (0, b + 4)) → D'
  6. Point(c) → G
  7. Segment(G, F_1) → e
  8. Segment(G, F_2) → f
  9. Translate(C, (b + 4, 0)) → C'_1
  10. Vector(C, C') → u
  11. Vector(D, D') → v
  12. Dilate(C, If[0 < n < 1, 1 - n, 0], C') → C'_2
  13. Dilate(D, If[0 < n < 1, 1 - n, 0], D') → D'_2
  14. Segment(C'_2, D'_2) → l
  15. Dilate(F_1, If[1 < n < 2, 2 - n, 1], C') → A'
  16. Vector(F_1, A') → w
  17. Translate(G, w) → G'
  18. Segment(A', G') → m
  19. Segment(C, D) → p
  20. Angle(F_2, F_1, G) → α
  21. Rotate(A', If[1 < n < 2, ((-(2 - n)) * α)], A') → A''
  22. Rotate(A', If[1 < n < 2, ((-(n - 2)) * α), 0], A'') → A''_1
  23. Rotate(A', If[1 < n < 2, ((-(n - 1)) * α), (-α)], A') → A''_2
  24. Rotate(G', If[1 < n < 2, ((-(n - 1)) * α), (-α)], A') → G''
  25. Segment(A''_2, G'') → m'
  26. Circle(C', e) → q
  27. Intersect(q, l, 1) → H
  28. Segment(C', H) → r
  29. Segment(F_1, G) → s
  30. Dilate(F_2, If[2 < n < 3, 3 - n, 1], D') → B'
  31. Vector(F_2, B') → t
  32. Translate(G, t) → G'_1
  33. Translate(G, t) → G'_2
  34. Translate(F_2, t) → B'_1
  35. Segment(G'_2, B'_1) → f'
  36. Angle(G, F_2, F_1) → β
  37. Segment(H, D') → a_1
  38. Segment(G, F_2) → b_1
  39. Center(c) → A
  40. Dilate(A, If[1 < n < 2, 2 - n, 0], C') → A'_1
  41. Vector(F_1, A'_1) → u_1
  42. Translate(A, u_1) → A'_2
  43. Translate(G, u_1) → G'_3
  44. Segment(A'_2, G'_3) → e_1
  45. Angle(D, F_1, G) → γ
  46. Rotate(A'_2, ((-(n - 1)) * γ), A'_2) → B
  47. Rotate(G'_3, ((-(n - 1)) * γ), A'_2) → I
  48. Dilate(G, If[2 < n < 3, 3 - n, 0], D') → G'_4
  49. Vector(G, G'_4) → v_1
  50. Translate(A, v_1) → A'_3
  51. Translate(G, v_1) → G'_5
  52. Segment(A'_3, G'_5) → g_1
  53. Rotate(A'_3, If[2 < n < 3, ((-(n - 2)) * γ)], G'_4) → J
  54. Rotate(G'_5, If[2 < n < 3, ((-(n - 2)) * γ)], G'_4) → K
  55. Circle(C', Distance[A''_2, G'']) → k_1
  56. Intersect(k_1, i_1, 2) → N
  57. Segment(C', N) → j_1
  58. Dilate(F_2, If[2 < n < 3, 3 - n], N) → O
  59. Vector(F_2, O) → w_1
  60. Translate(G, w_1) → G'_6
  61. Translate(F_2, w_1) → L
  62. Segment(G'_6, L) → f'_1
  63. Circle(N, f) → p_1
  64. Rotate(G'_6, If[2 < n < 3, ((n - 2) * β)], O) → M
  65. Rotate(L, If[2 < n < 3, ((n - 2) * β)], O) → L'
  66. Segment(M, L') → l_1
  67. Segment(N, D') → m_1
  68. Dilate(C', If[3 < n ≤ 4, 4 - n], C) → C''
  69. Dilate(D', If[3 < n ≤ 4, 4 - n], D) → D''
  70. Segment(C'', D'') → n_1

AI Drawing Prompt Reference

Draw a hyperbola, marking the foci and vertices, and showing the difference in focal distances. Start by plotting the vertices and foci, then use translation and dilation to construct the asymptotes, and finally connect key points to form the complete shape, labeling angles and distances as needed.
The prompt above was compiled from the original drawing commands and is for reference only.