11X1 T12 05 tangents & normals I

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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) Tangentyx 2 4ay x 4. Tangents &amp; Normals (i) Using Parametrics (1) Tangentyx 2 4ayP(2ap, ap 2 ) x 5. Tangents &amp; Normals (i) Using Parametrics (1) Tangentyx 2 4ayP(2ap, ap 2 ) x 6. Tangents &amp; Normals (i) Using Parametrics (1) Tangentyx 2 4ayP(2ap, ap 2 ) xx2y 4a 7. Tangents &amp; Normals (i) Using Parametrics (1) Tangentyx 2 4ayP(2ap, ap 2 ) xx2y 4a dy x dx 2a 8. Tangents &amp; Normals(i) Using Parametrics (1) Tangenty x 2 4ay P(2ap, ap 2 )xx2y ORx 2 4ay 4a dy x dx 2a 9. Tangents &amp; Normals(i) Using Parametrics (1) Tangenty x 2 4ay P(2ap, ap 2 )xx2y ORx 2 4ay 4ady dy x2 x 4a dx dx 2a 10. Tangents &amp; Normals(i) Using Parametrics (1) Tangenty x 2 4ay P(2ap, ap 2 )xx2y ORx 2 4ay 4ady dy x2 x 4a dx dx 2a dy x dx 2a 11. Tangents &amp; Normals(i) Using Parametrics (1) Tangenty x 2 4ay P(2ap, ap 2 )xx2y ORx 2 4ay 4ady dy x2 x 4a dx dx 2a dy x dy dx 2a when x 2ap, pdx 12. Tangents &amp; Normals(i) Using Parametrics (1) TangentORy x 2 4ay x 2at P(2ap, ap 2 )xx2y ORx 2 4ay 4ady dy x2 x 4a dx dx 2a dy x dy dx 2a when x 2ap, pdx 13. Tangents &amp; Normals(i) Using Parametrics (1) TangentORy x 2 4ay x 2at dx 2a dt P(2ap, ap 2 )xx2y ORx 2 4ay 4ady dy x2 x 4a dx dx 2a dy x dy dx 2a when x 2ap, pdx 14. Tangents &amp; Normals(i) Using Parametrics (1) TangentORy x 2 4ay x 2aty at 2 dx 2a dt P(2ap, ap 2 )xx2y ORx 2 4ay 4ady dy x2 x 4a dx dx 2a dy x dy dx 2a when x 2ap, pdx 15. Tangents &amp; Normals(i) Using Parametrics (1) TangentORy x 2 4ay x 2aty at 2 dxdy 2a 2at dtdt P(2ap, ap 2 )xx2y ORx 2 4ay 4ady dy x2 x 4a dx dx 2a dy x dy dx 2a when x 2ap, pdx 16. Tangents &amp; Normals (i) Using Parametrics (1) Tangent ORyx 2 4ay x 2aty at 2dxdy 2a 2atdtdtdy dy dt2P(2ap, ap ) dx dt dxxx2y ORx 2 4ay 4ady dy x2 x 4a dx dx 2a dy x dy dx 2a when x 2ap, pdx 17. Tangents &amp; Normals(i) Using Parametrics (1) Tangent ORy x 2 4ay x 2at y at 2dx dy 2a 2atdt dtdy dy dt 2 P(2ap, ap ) dx dt dx dy 1 x 2at dx 2a x2 t yORx 4ay 2 4a dydy x 2 x 4a dxdx 2ady x dy dx 2awhen x 2ap, p dx 18. Tangents &amp; Normals(i) Using Parametrics (1) Tangent ORy x 2 4ay x 2at y at 2dx dy 2a 2atdt dtdy dy dt 2 P(2ap, ap ) dx dt dx dy 1 dy x 2at when t p, p dx 2adx x2 t yORx 4ay 2 4a dydy x 2 x 4a dxdx 2ady x dy dx 2awhen x 2ap, p dx 19. Tangents &amp; Normals(i) Using Parametrics (1) Tangent ORy x 2 4ay x 2at y at 2dx dy 2a 2atdt dtdy dy dt 2 P(2ap, ap ) dx dt dx dy 1dy x 2at when t p, p dx 2a dx x2 t slope of tangent is p yORx 4ay 2 4a dydy x 2 x 4a dxdx 2ady x dy dx 2awhen x 2ap, p dx 20. Tangents &amp; Normals(i) Using Parametrics (1) Tangent ORy x 2 4ay x 2at y at 2dx dy 2a 2atdt dtdy dy dt 2 P(2ap, ap ) dx dt dx dy 1 dy x 2at when t p, p dx 2adx x2 t slope of tangent is p yORx 4ay 2 4a dy y ap 2 p x 2ap dy x 2 x 4a dxdx 2ady x dy dx 2awhen x 2ap, p dx 21. Tangents &amp; Normals(i) Using Parametrics (1) Tangent ORy x 2 4ay x 2at y at 2dx dy 2a 2atdt dtdy dy dt 2 P(2ap, ap ) dx dt dx dy 1 dy x 2at when t p, p dx 2adx x2 t slope of tangent is p yORx 4ay 2 4a dy y ap 2 p x 2ap dy x 2 x 4a dxdx 2ady x y ap 2 px 2ap 2 dy dx 2awhen x 2ap, p dx 22. Tangents &amp; Normals(i) Using Parametrics (1) Tangent ORy x 2 4ay x 2at y at 2dx dy 2a 2atdt dtdy dy dt 2 P(2ap, ap ) dx dt dx dy 1 dy x 2at when t p, p dx 2adx x2 t slope of tangent is p yORx 4ay 2 4a dy y ap 2 p x 2ap dy x 2 x 4a dxdx 2ady x y ap 2 px 2ap 2 dydx 2a y px ap 2when x 2ap, p dx 23. (2) Tangents from an external point 24. (2) Tangents from an external pointx2 4 y yx 25. (2) Tangents from an external pointx2 4 y y(3, 2) x 26. (2) Tangents from an external pointx2 4 y y(3, 2) x 27. (2) Tangents from an external pointx2 4 y y(3, 2)P(2 p, p 2 ) x 28. (2) Tangents from an external pointx2 4 y y(3, 2)P(2 p, p 2 ) x 29. (2) Tangents from an external pointx2 4 y yQ(2q, q 2 ) (3, 2)P(2 p, p 2 ) x 30. (2) Tangents from an external pointx2x2 4 y y y4Q(2q, q 2 ) (3, 2)P(2 p, p 2 ) x 31. (2) Tangents from an external point x2x2 4 yy y 4Q(2q, q 2 )dy x dx 2(3, 2)P(2 p, p 2 ) x 32. (2) Tangents from an external pointx2x2 4 y y y4 Q(2q, q 2 )dy xdx 2dy when x 2 p, p(3, 2)dx P(2 p, p 2 )x 33. (2) Tangents from an external point x2x2 4 yy y 4 Q(2q, q 2 ) dy x dx 2dy when x 2 p, p(3, 2)dx P(2 p, p 2 ) y p2 p x 2 p x 34. (2) Tangents from an external pointx2x2 4 y y y4 Q(2q, q 2 )dy xdx 2dy when x 2 p, p(3, 2)dx P(2 p, p 2 ) y p2 p x 2 p xy p 2 px 2 p 2 35. (2) Tangents from an external pointx2x2 4 y y y4 Q(2q, q 2 )dy xdx 2dy when x 2 p, p(3, 2)dx P(2 p, p 2 ) y p2 p x 2 p xy p 2 px 2 p 2 y px p 2 36. (2) Tangents from an external pointx2x2 4 y y y4 Q(2q, q 2 )dy xdx 2dy when x 2 p, p(3, 2)dx P(2 p, p 2 ) y p2 p x 2 p xy p 2 px 2 p 2 y px p 2 however tangent passes through (3,2) 37. (2) Tangents from an external pointx2x2 4 y y y4 Q(2q, q 2 )dy xdx 2dy when x 2 p, p(3, 2)dx P(2 p, p 2 ) y p2 p x 2 p xy p 2 px 2 p 2 y px p 2 however tangent passes through (3,2) 2 3 p p2 38. (2) Tangents from an external pointx2x2 4 y y y4 Q(2q, q 2 )dy xdx 2dy when x 2 p, p(3, 2)dx P(2 p, p 2 ) y p2 p x 2 p xy p 2 px 2 p 2 y px p 2 however tangent passes through (3,2) 2 3 p p2p2 3 p 2 0 39. (2) Tangents from an external pointx2x2 4 y y y4 Q(2q, q 2 )dy xdx 2dy when x 2 p, p(3, 2)dx P(2 p, p 2 ) y p2 p x 2 p xy p 2 px 2 p 2y px p 2 however tangent passes through (3,2)2 3 p p2p2 3 p 2 0 p 2 p 1 0 40. (2) Tangents from an external pointx2x2 4 y y y4 Q(2q, q 2 )dy xdx 2dy when x 2 p, p(3, 2)dx P(2 p, p 2 ) y p2 p x 2 p xy p 2 px 2 p 2y px p 2 however tangent passes through (3,2)2 3 p p2p2 3 p 2 0 p 2 p 1 0p 1 or p 2 41. (2) Tangents from an external pointx2x2 4 y y y4 Q(2q, q 2 )dy xdx 2dy when x 2 p, p(3, 2)dx P(2 p, p 2 ) y p2 p x 2 p xy p 2 px 2 p 2y px p 2 however tangent passes through (3,2)2 3 p p2p2 3 p 2 0 p 2 p 1 0p 1 or p 2 tangents are y x 1 and y 2 x 4 42. (3) Intersection of tangents 43. (3) Intersection of tangentsy x 2 4ayx 44. (3) Intersection of tangentsy x 2 4ay P(2ap, ap 2 )x 45. (3) Intersection of tangentsy x 2 4ay P(2ap, ap 2 )x 46. (3) Intersection of tangentsy x 2 4ay1 Show tangent at P is y px ap 2P(2ap, ap 2 )x 47. (3) Intersection of tangents y x 2 4ay 1 Show tangent at P is y px ap 2 Q(2aq, aq 2 ) P(2ap, ap 2 ) x 48. (3) Intersection of tangents y 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 tangents y 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 tangents y 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 T 51. (3) Intersection of tangents y 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 T 52. (3) Intersection of tangents y 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 2 T qx y aq 2 53. (3) Intersection of tangents y 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 2 T qx y aq 2 p q x a p 2 q 2 54. (3) Intersection of tangents y 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 2 T 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 tangents y 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 2 T 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 tangents y 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 2 T 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 y ap 2 apq ap 2 apq 57. (3) Intersection of tangents y 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 2 T 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 y ap 2 apq ap 2 apq T a p q , apq 58. (4) Normal 59. (4) Normal yx 2 4ayx 60. (4) Normal yx 2 4ay P(2ap, ap 2 )x 61. (4) Normal yx 2 4ay P(2ap, ap 2 )x 62. (4) Normal yx 2 4ay P(2ap, ap 2 ) 1 Show the slope of tangent at P is p x 63. (4) Normal yx 2 4ay P(2ap, ap 2 ) 1 Show the slope of tangent at P is p 1x 2 slope of normal is p 64. (4) Normal yx 2 4ay P(2ap, ap 2 ) 1 Show the slope of tangent at P is p 1x 2 slope of normal is p 1 y ap x 2ap 2 p 65. (4) Normal yx 2 4ay P(2ap, ap 2 ) 1 Show the slope of tangent at P is p 1x 2 slope of normal is p 1 y ap x 2ap 2 ppy ap 3 x 2ap 66. (4) Normal yx 2 4ay P(2ap, ap 2 ) 1 Show the slope of tangent at P is p 1x 2 slope of normal is p 1 y ap x 2ap 2 ppy ap 3 x 2apx py ap 3 2ap 67. (5) Intersection of normals 68. (5) Intersection of normalsyx 2 4ay x 69. (5) Intersection of normals y x 2 4ayP(2ap, ap 2 )x 70. (5) Intersection of normalsyx 2 4ayP(2ap, ap 2 ) x 71. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3yP(2ap, ap 2 ) x 72. (5) Intersection of normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3y Q(2aq, aq 2 )P(2ap, ap 2 )x 73. (5) Intersection of normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3y Q(2aq, aq 2 )P(2ap, ap 2 )x 74. (5) Intersection of normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3y2 normal at Q is x qy aq 2aq3Q(2aq, aq 2 )P(2ap, ap 2 )x 75. (5) Intersection of normals x 2 4ay 1 Show normal at P is x py ap 2ap3y 2 normal at Q is x qy aq 2aq 3N Q(2aq, aq 2 ) P(2ap, ap 2 ) x 76. (5) Intersection of normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3y2 normal at Q is x qy aq 2aq3N Q(2aq, aq 2 )2 3 Solve simultaneously P(2ap, ap )x 77. (5) Intersection of normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3y 2 normal at Q is x qy aq 2aq 3N Q(2aq, aq 2 )23 Solve simultaneously P(2ap, ap )x py ap 3 2apxx qy aq 3 2aq 78. (5) Intersection of normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3y2 normal at Q is x qy aq 2aq3N Q(2aq, aq 2 )2 3 Solve simultaneously P(2ap, ap )x py ap 3 2apxx qy aq 3 2aq p q y a p 3 q 3 2a p q 79. (5) Intersection of normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3y2 normal at Q is x qy aq 2aq3N Q(2aq, aq 2 )2 3 Solve simultaneously P(2ap, ap )x py ap 3 2apxx 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 y2 normal at Q is x qy aq 2aq3N Q(2aq, aq 2 )2 3 Solve simultaneouslyP(2ap, ap )x py ap 3 2apxx 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 x ap p 2 pq q 2 2 ap 3 2ap 81. (5) Intersection of normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3y2 normal at Q is x qy aq 2aq3N Q(2aq, aq 2 )2 3 Solve simultaneously P(2ap, ap )x py ap 3 2apxx 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 x 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 normalsx 2 4ay 1 Show normal at P is x py ap 2ap 3y2 normal at Q is x qy aq 2aq3N Q(2aq, aq 2 )2 3 Solve simultaneously P(2ap, ap )x py ap 3 2apxx 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 x 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>