11 x1 t11 05 tangents & normals i (2012)

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<ul><li> 1. Tangents &amp; Normals (i) Using Parametrics</li></ul> <p> 2. Tangents &amp; Normals(i) Using Parametrics(1) Tangent 3. Tangents &amp; Normals (i) Using Parametrics(1) Tangent yx 2 4ay x 4. Tangents &amp; Normals(i) Using Parametrics(1) Tangent yx 2 4ayP(2ap, ap 2 )x 5. Tangents &amp; Normals(i) Using Parametrics(1) Tangent yx 2 4ayP(2ap, ap 2 )x 6. Tangents &amp; Normals(i) Using Parametrics(1) Tangent yx 2 4ayP(2ap, ap 2 )x x2y 4a 7. Tangents &amp; Normals(i) Using Parametrics(1) Tangent yx 2 4ayP(2ap, ap 2 )xx2 y4ady xdx 2a 8. Tangents &amp; Normals (i) Using Parametrics(1) Tangent y x 2 4ay P(2ap, ap 2 ) xx2 y ORx 2 4ay4ady xdx 2a 9. Tangents &amp; Normals (i) Using Parametrics(1) Tangent y x 2 4ay P(2ap, ap 2 ) xx2 y ORx 2 4ay4a dydy x2 x 4adxdx 2a 10. Tangents &amp; Normals (i) Using Parametrics(1) Tangent y x 2 4ay P(2ap, ap 2 ) xx2 y ORx 2 4ay4a dydy x2 x 4adxdx 2a dy xdx 2a 11. Tangents &amp; Normals (i) Using Parametrics(1) Tangent y x 2 4ay P(2ap, ap 2 ) xx2 y ORx 2 4ay4a dydy x2 x 4adxdx 2a dy x dy dx 2awhen x 2ap, p dx 12. Tangents &amp; Normals (i) Using Parametrics(1) TangentOR y x 2 4ay x 2at P(2ap, ap 2 ) xx2 y ORx 2 4ay4a dydy x2 x 4adxdx 2a dy x dy dx 2awhen x 2ap, p dx 13. Tangents &amp; Normals (i) Using Parametrics(1) TangentOR y x 2 4ay x 2atdx 2adt P(2ap, ap 2 ) xx2 y ORx 2 4ay4a dydy x2 x 4adxdx 2a dy x dy dx 2awhen x 2ap, p dx 14. Tangents &amp; Normals (i) Using Parametrics(1) TangentOR y x 2 4ay x 2aty at 2dx 2adt P(2ap, ap 2 ) xx2 y ORx 2 4ay4a dydy x2 x 4adxdx 2a dy x dy dx 2awhen x 2ap, p dx 15. Tangents &amp; Normals (i) Using Parametrics(1) TangentOR y x 2 4ay x 2aty at 2dxdy 2a 2atdtdt P(2ap, ap 2 ) xx2 y ORx 2 4ay4a dydy x2 x 4adxdx 2a dy x dy dx 2awhen x 2ap, p dx 16. Tangents &amp; Normals(i) Using Parametrics(1) Tangent OR yx 2 4ay x 2aty at 2 dxdy 2a 2at dtdt dy dy dt 2 P(2ap, ap ) dx dt dx xx2 y ORx 2 4ay4a dydy x2 x 4adxdx 2a dy x dy dx 2awhen x 2ap, p dx 17. Tangents &amp; Normals (i) Using Parametrics(1) Tangent OR y x 2 4ay x 2at y at 2 dx dy 2a 2at dt dt dy dy dt2P(2ap, ap ) dx dt dxdy1x 2at dx 2ax2 tyORx 4ay24ady dy x 2 x 4a dx dx 2ady xdy dx 2a when x 2ap, pdx 18. Tangents &amp; Normals (i) Using Parametrics(1) Tangent OR y x 2 4ay x 2at y at 2 dx dy 2a 2at dt dt dy dy dt2P(2ap, ap ) dx dt dxdy1dyx 2at when t p, pdx 2adxx2 tyORx 4ay24ady dy x 2 x 4a dx dx 2ady xdy dx 2a when x 2ap, pdx 19. Tangents &amp; Normals (i) Using Parametrics(1) Tangent OR y x 2 4ay x 2at y at 2 dx dy 2a 2at dt dt dy dy dt2P(2ap, ap ) dx dt dxdy1dyx 2at when t p, pdx 2adxx2 t slope of tangent is pyORx 4ay24ady dy x 2 x 4a dx dx 2ady xdy dx 2a when x 2ap, pdx 20. Tangents &amp; Normals (i) Using Parametrics(1) Tangent OR y x 2 4ay x 2at y at 2 dx dy 2a 2at dt dt dy dy dt2P(2ap, ap ) dx dt dxdy1dyx 2at when t p, pdx 2adxx2 t slope of tangent is pyORx 4ay24ady y ap 2 p x 2ap dy x 2 x 4a dx dx 2ady xdy dx 2a when x 2ap, pdx 21. Tangents &amp; Normals (i) Using Parametrics(1) Tangent OR y x 2 4ay x 2at y at 2 dx dy 2a 2at dt dt dy dy dt2P(2ap, ap ) dx dt dxdy1dyx 2at when t p, pdx 2adxx2 t slope of tangent is pyORx 4ay24ady y ap 2 p x 2ap dy x 2 x 4a dx dx 2ady x y ap 2 px 2ap 2dy dx 2a when x 2ap, pdx 22. Tangents &amp; Normals (i) Using Parametrics(1) Tangent OR y x 2 4ay x 2at y at 2 dx dy 2a 2at dt dt dy dy dt2P(2ap, ap ) dx dt dxdy1dyx 2at when t p, pdx 2adxx2 t slope of tangent is pyORx 4ay24ady y ap 2 p x 2ap dy x 2 x 4a dx dx 2ady x y ap 2 px 2ap 2dydx 2a y px ap 2 when x 2ap, pdx 23. (2) Tangents from an external point 24. (2) Tangents from an external point x2 4 y y x 25. (2) Tangents from an external point x2 4 y y(3, 2) x 26. (2) Tangents from an external point x2 4 y y(3, 2) x 27. (2) Tangents from an external point x2 4 y y(3, 2) P(2 p, p 2 )x 28. (2) Tangents from an external point x2 4 y y(3, 2) P(2 p, p 2 )x 29. (2) Tangents from an external point x2 4 y yQ(2q, q 2 )(3, 2) P(2 p, p 2 )x 30. (2) Tangents from an external pointx2 x2 4 y y y 4Q(2q, q 2 )(3, 2) P(2 p, p 2 )x 31. (2) Tangents from an external point x2 x2 4 yy y4Q(2q, q 2 ) dy xdx 2(3, 2) P(2 p, p 2 )x 32. (2) Tangents from an external pointx2 x2 4 y y y 4Q(2q, q 2 )dy x dx 2 dywhen x 2 p, p (3, 2)dxP(2 p, p 2 ) x 33. (2) Tangents from an external point x2 x2 4 yy y4Q(2q, q 2 ) dy xdx 2 dywhen x 2 p, p (3, 2)dxP(2 p, p 2 ) y p2 p x 2 p x 34. (2) Tangents from an external pointx2 x2 4 y y y 4Q(2q, q 2 )dy x dx 2 dywhen x 2 p, p (3, 2)dxP(2 p, p 2 ) y p2 p x 2 p x y p 2 px 2 p 2 35. (2) Tangents from an external pointx2 x2 4 y y y 4Q(2q, q 2 )dy x dx 2 dywhen x 2 p, p (3, 2)dxP(2 p, p 2 ) y p2 p x 2 p x y p 2 px 2 p 2y px p 2 36. (2) Tangents from an external pointx2 x2 4 y y y 4Q(2q, q 2 )dy x dx 2 dywhen x 2 p, p (3, 2)dxP(2 p, p 2 ) y p2 p x 2 p x y p 2 px 2 p 2y px p 2however tangent passes through (3,2) 37. (2) Tangents from an external pointx2 x2 4 y y y 4Q(2q, q 2 )dy x dx 2 dywhen x 2 p, p (3, 2)dxP(2 p, p 2 ) y p2 p x 2 p x y p 2 px 2 p 2y px p 2however tangent passes through (3,2)2 3p p 2 38. (2) Tangents from an external pointx2 x2 4 y y y 4Q(2q, q 2 )dy x dx 2 dywhen x 2 p, p (3, 2)dxP(2 p, p 2 ) y p2 p x 2 p x y p 2 px 2 p 2y px p 2however tangent passes through (3,2)2 3p p 2 p2 3 p 2 0 39. (2) Tangents from an external pointx2 x2 4 y y y 4Q(2q, q 2 )dy x dx 2 dywhen x 2 p, p (3, 2)dxP(2 p, p 2 ) y p2 p x 2 p x y p 2 px 2 p 2 y px p 2however tangent passes through (3,2) 2 3p p 2 p2 3 p 2 0 p 2 p 1 0 40. (2) Tangents from an external pointx2 x2 4 y y y 4Q(2q, q 2 )dy x dx 2 dywhen x 2 p, p (3, 2)dxP(2 p, p 2 ) y p2 p x 2 p x y p 2 px 2 p 2 y px p 2however tangent passes through (3,2) 2 3p p 2 p2 3 p 2 0 p 2 p 1 0 p 1 or p 2 41. (2) Tangents from an external pointx2 x2 4 y y y 4Q(2q, q 2 )dy x dx 2 dywhen x 2 p, p (3, 2)dxP(2 p, p 2 ) y p2 p x 2 p x y p 2 px 2 p 2 y px p 2however tangent passes through (3,2) 2 3p p 2 p2 3 p 2 0 p 2 p 1 0 p 1 or p 2 tangents are y x 1 and y 2 x 4 42. (3) Intersection of tangents 43. (3) Intersection of tangents y x 2 4ay x 44. (3) Intersection of tangents y x 2 4ay P(2ap, ap 2 ) x 45. (3) Intersection of tangents y x 2 4ay P(2ap, ap 2 ) x 46. (3) Intersection of tangents y x 2 4ay 1 Show tangent at P is y px ap 2 P(2ap, ap 2 ) x 47. (3) Intersection of tangentsy x 2 4ay1 Show tangent at P is y px ap 2Q(2aq, aq 2 ) P(2ap, ap 2 )x 48. (3) Intersection of tangentsy x 2 4ay 1 Show tangent at P is y px ap 2Q(2aq, aq 2 )P(2ap, ap 2 ) x 49. (3) Intersection of tangentsy x 2 4ay 1 Show tangent at P is y px ap 2 2 tangent at Q is y qx aq 2Q(2aq, aq 2 )P(2ap, ap 2 ) x 50. (3) Intersection of tangentsy x 2 4ay 1 Show tangent at P is y px ap 2 2 tangent at Q is y qx aq 2Q(2aq, aq 2 )P(2ap, ap 2 ) xT 51. (3) Intersection of tangentsy x 2 4ay 1 Show tangent at P is y px ap 2 2 tangent at Q is y qx aq 2Q(2aq, aq 2 )P(2ap, ap 2 ) 3 Solve simultaneously xT 52. (3) Intersection of tangentsy x 2 4ay 1 Show tangent at P is y px ap 2 2 tangent at Q is y qx aq 2Q(2aq, aq 2 )P(2ap, ap 2 ) 3 Solve simultaneously xpx y ap 2T qx y aq 2 53. (3) Intersection of tangentsy x 2 4ay 1 Show tangent at P is y px ap 2 2 tangent at Q is y qx aq 2Q(2aq, aq 2 )P(2ap, ap 2 ) 3 Solve simultaneously x px y ap 2T qx y aq 2 p q x a p 2 q 2 54. (3) Intersection of tangentsy x 2 4ay 1 Show tangent at P is y px ap 2 2 tangent at Q is y qx aq 2Q(2aq, aq 2 )P(2ap, ap 2 ) 3 Solve simultaneously xpx y ap 2T qx y aq 2 p q x a p 2 q 2 p q x a p q p q x a p q 55. (3) Intersection of tangentsy x 2 4ay 1 Show tangent at P is y px ap 2 2 tangent at Q is y qx aq 2Q(2aq, aq 2 )P(2ap, ap 2 ) 3 Solve simultaneously xpx y ap 2T qx y aq 2 p q x a p 2 q 2 p q x a p q p q x a p q y ap p q ap 2 56. (3) Intersection of tangentsy x 2 4ay 1 Show tangent at P is y px ap 2 2 tangent at Q is y qx aq 2Q(2aq, aq 2 )P(2ap, ap 2 ) 3 Solve simultaneously xpx y ap 2T qx y aq 2 p q x a p 2 q 2 p q x a p q p q x a p q y ap p q ap 2y ap 2 apq ap 2 apq 57. (3) Intersection of tangentsy x 2 4ay 1 Show tangent at P is y px ap 2 2 tangent at Q is y qx aq 2Q(2aq, aq 2 )P(2ap, ap 2 ) 3 Solve simultaneously xpx y ap 2T qx y aq 2 p q x a p 2 q 2 p q x a p q p q x a p q y ap p q ap 2y ap 2 apq ap 2 apqT a p q , apq 58. (4) Normal 59. (4) Normalyx 2 4ayx 60. (4) Normalyx 2 4ay P(2ap, ap 2 ) x 61. (4) Normalyx 2 4ay P(2ap, ap 2 ) x 62. (4) Normalyx 2 4ay P(2ap, ap 2 ) 1 Show the slope of tangent at P is p x 63. (4) Normalyx 2 4ay P(2ap, ap 2 ) 1 Show the slope of tangent at P is p1 x 2 slope of normal is p 64. (4) Normalyx 2 4ay P(2ap, ap 2 ) 1 Show the slope of tangent at P is p1 x 2 slope of normal is p1y ap x 2ap 2 p 65. (4) Normalyx 2 4ay P(2ap, ap 2 ) 1 Show the slope of tangent at P is p1 x 2 slope of normal is p1y ap x 2ap 2 p py ap 3 x 2ap 66. (4) Normalyx 2 4ay P(2ap, ap 2 ) 1 Show the slope of tangent at P is p1 x 2 slope of normal is p1y ap x 2ap 2 p py ap 3 x 2ap x py ap 3 2ap 67. (5) Intersection of normals 68. (5) Intersection of normals yx 2 4ay x 69. (5) Intersection of normalsy x 2 4ayP(2ap, ap 2 ) x 70. (5) Intersection of normals yx 2 4ayP(2ap, ap 2 )x 71. (5) Intersection of normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3 yP(2ap, ap 2 )x 72. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3 yQ(2aq, aq 2 ) P(2ap, ap 2 ) x 73. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3 yQ(2aq, aq 2 ) P(2ap, ap 2 ) x 74. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3 y 2 normal at Q is x qy aq 2aq 3Q(2aq, aq 2 ) P(2ap, ap 2 ) x 75. (5) Intersection of normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3 y2 normal at Q is x qy aq 2aq3NQ(2aq, aq 2 )P(2ap, ap 2 )x 76. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3 y 2 normal at Q is x qy aq 2aq 3NQ(2aq, aq 2 )2 3 Solve simultaneouslyP(2ap, ap ) x 77. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3 y2 normal at Q is x qy aq 2aq3NQ(2aq, aq 2 )23 Solve simultaneouslyP(2ap, ap ) x py ap 3 2ap x x qy aq 3 2aq 78. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3 y 2 normal at Q is x qy aq 2aq 3NQ(2aq, aq 2 )2 3 Solve simultaneouslyP(2ap, ap ) x py ap 3 2ap x x qy aq 3 2aq p q y a p 3 q 3 2a p q 79. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3 y 2 normal at Q is x qy aq 2aq 3NQ(2aq, aq 2 )2 3 Solve simultaneouslyP(2ap, ap ) x py ap 3 2ap x x qy aq 3 2aq p q y a p 3 q 3 2a p q p q y a p q p 2 pq q 2 2a p q y a p 2 pq q 2 2 80. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3 y 2 normal at Q is x qy aq 2aq 3NQ(2aq, aq 2 )2 3 Solve simultaneouslyP(2ap, ap ) x py ap 3 2ap x x qy aq 3 2aq p q y a p 3 q 3 2a p q p q y a p q p 2 pq q 2 2a p q y a p 2 pq q 2 2x ap p 2 pq q 2 2 ap 3 2ap 81. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3 y 2 normal at Q is x qy aq 2aq 3NQ(2aq, aq 2 )2 3 Solve simultaneouslyP(2ap, ap ) x py ap 3 2ap x x qy aq 3 2aq p q y a p 3 q 3 2a p q p q y a p q p 2 pq q 2 2a p q y a p 2 pq q 2 2x ap p 2 pq q 2 2 ap 3 2apx ap 3 2ap ap 3 ap 2 q apq 2 2ap apq p q 82. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3 y 2 normal at Q is x qy aq 2aq 3NQ(2aq, aq 2 )2 3 Solve simultaneouslyP(2ap, ap ) x py ap 3 2ap x x qy aq 3 2aq p q y a p 3 q 3 2a p q p q y a p q p 2 pq q 2 2a p q y a p 2 pq q 2 2x ap p 2 pq q 2 2 ap 3 2apx ap 3 2ap ap 3 ap 2 q apq 2 2ap apq p q N apq p q , a p 2 pq q 2 2 83. Exercise 9F; 1ac, 2ac, 3, 6, 8, 9ac, 11, 12, 13</p>