1. Tangents & Normals
(i) Using Parametrics

2. Tangents & Normals
(i) Using Parametrics
(1) Tangent

3. Tangents & Normals
(i) Using Parametrics
(1) Tangent
y x 2  4ay
x

4. Tangents & Normals
(i) Using Parametrics
(1) Tangent
y x 2  4ay
P(2ap, ap 2 )
x

5. Tangents & Normals
(i) Using Parametrics
(1) Tangent
y x 2  4ay
P(2ap, ap 2 )
x

6. Tangents & Normals
(i) Using Parametrics
(1) Tangent
y x 2  4ay
P(2ap, ap 2 )
x
x2
y
4a

7. Tangents & Normals
(i) Using Parametrics
(1) Tangent
y x 2  4ay
P(2ap, ap 2 )
x
x2
y
4a
dy x

dx 2a

8. Tangents & Normals
(i) Using Parametrics
(1) Tangent
y x 2  4ay
P(2ap, ap 2 )
x
x2
y OR x 2  4ay
4a
dy x

dx 2a

9. Tangents & Normals
(i) Using Parametrics
(1) Tangent
y x 2  4ay
P(2ap, ap 2 )
x
x2
y OR x 2  4ay
4a
dy
dy x 2 x  4a
 dx
dx 2a

10. Tangents & Normals
(i) Using Parametrics
(1) Tangent
y x 2  4ay
P(2ap, ap 2 )
x
x2
y OR x 2  4ay
4a
dy
dy x 2 x  4a
 dx
dx 2a dy x

dx 2a

11. Tangents & Normals
(i) Using Parametrics
(1) Tangent
y x 2  4ay
P(2ap, ap 2 )
x
x2
y ORx 2  4ay
4a
dy
dy x 2 x  4a
 dx
dx 2a dy x

dy dx 2a
when x  2ap,  p
dx

12. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at
P(2ap, ap 2 )
x
x2
y ORx 2  4ay
4a
dy
dy x 2 x  4a
 dx
dx 2a dy x

dy dx 2a
when x  2ap,  p
dx

13. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at
dx
 2a
dt
P(2ap, ap 2 )
x
x2
y ORx 2  4ay
4a
dy
dy x 2 x  4a
 dx
dx 2a dy x

dy dx 2a
when x  2ap,  p
dx

14. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at y  at 2
dx
 2a
dt
P(2ap, ap 2 )
x
x2
y ORx 2  4ay
4a
dy
dy x 2 x  4a
 dx
dx 2a dy x

dy dx 2a
when x  2ap,  p
dx

15. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at y  at 2
dx dy
 2a  2at
dt dt
P(2ap, ap 2 )
x
x2
y ORx 2  4ay
4a
dy
dy x 2 x  4a
 dx
dx 2a dy x

dy dx 2a
when x  2ap,  p
dx

16. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at y  at 2
dx dy
 2a  2at
dt dt
dy dy dt
2
P(2ap, ap )  
dx dt dx
x
x2
y ORx 2  4ay
4a
dy
dy x 2 x  4a
 dx
dx 2a dy x

dy dx 2a
when x  2ap,  p
dx

17. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at y  at 2
dx dy
 2a  2at
dt dt
dy dy dt
2
P(2ap, ap )  
dx dt dx
dy 1
x  2at 
dx 2a
x2 t
y OR x  4ay
2
4a
dy
dy x 2 x  4a
 dx
dx 2a dy x

dy dx 2a
when x  2ap,  p
dx

18. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at y  at 2
dx dy
 2a  2at
dt dt
dy dy dt
2
P(2ap, ap )  
dx dt dx
dy 1 dy
x  2at  when t  p,  p
dx 2a dx
x2 t
y OR x  4ay
2
4a
dy
dy x 2 x  4a
 dx
dx 2a dy x

dy dx 2a
when x  2ap,  p
dx

19. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at y  at 2
dx dy
 2a  2at
dt dt
dy dy dt
2
P(2ap, ap )  
dx dt dx
dy 1 dy
x  2at  when t  p,  p
dx 2a dx
x2 t  slope of tangent is p
y OR x  4ay
2
4a
dy
dy x 2 x  4a
 dx
dx 2a dy x

dy dx 2a
when x  2ap,  p
dx

20. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at y  at 2
dx dy
 2a  2at
dt dt
dy dy dt
2
P(2ap, ap )  
dx dt dx
dy 1 dy
x  2at  when t  p,  p
dx 2a dx
x2 t  slope of tangent is p
y OR x  4ay
2
4a
dy y  ap 2  p x  2ap 
dy x 2 x  4a
 dx
dx 2a dy x

dy dx 2a
when x  2ap,  p
dx

21. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at y  at 2
dx dy
 2a  2at
dt dt
dy dy dt
2
P(2ap, ap )  
dx dt dx
dy 1 dy
x  2at  when t  p,  p
dx 2a dx
x2 t  slope of tangent is p
y OR x  4ay
2
4a
dy y  ap 2  p x  2ap 
dy x 2 x  4a
 dx
dx 2a dy x y  ap 2  px  2ap 2

dy dx 2a
when x  2ap,  p
dx

22. Tangents & Normals
(i) Using Parametrics
(1) Tangent OR
y x 2  4ay x  2at y  at 2
dx dy
 2a  2at
dt dt
dy dy dt
2
P(2ap, ap )  
dx dt dx
dy 1 dy
x  2at  when t  p,  p
dx 2a dx
x2 t  slope of tangent is p
y OR x  4ay
2
4a
dy y  ap 2  p x  2ap 
dy x 2 x  4a
 dx
dx 2a dy x y  ap 2  px  2ap 2

dy dx 2a y  px  ap 2
when x  2ap,  p
dx

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
y Q(2q, q 2 )
(3, 2)
P(2 p, p 2 )
x

30. (2) Tangents from an external point x2
x2  4 y y
y 4
Q(2q, q 2 )
(3, 2)
P(2 p, p 2 )
x

31. (2) Tangents from an external point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
(3, 2)
P(2 p, p 2 )
x

32. (2) Tangents from an external point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
dy
when x  2 p,  p
(3, 2) dx
P(2 p, p 2 )
x

33. (2) Tangents from an external point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
dy
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 point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
dy
when x  2 p,  p
(3, 2) dx
P(2 p, p 2 ) y  p2  p  x  2 p 
x
y  p 2  px  2 p 2

35. (2) Tangents from an external point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
dy
when x  2 p,  p
(3, 2) dx
P(2 p, p 2 ) y  p2  p  x  2 p 
x
y  p 2  px  2 p 2
y  px  p 2

36. (2) Tangents from an external point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
dy
when x  2 p,  p
(3, 2) dx
P(2 p, p 2 ) y  p2  p  x  2 p 
x
y  p 2  px  2 p 2
y  px  p 2
however tangent passes through (3,2)

37. (2) Tangents from an external point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
dy
when x  2 p,  p
(3, 2) dx
P(2 p, p 2 ) y  p2  p  x  2 p 
x
y  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 point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
dy
when x  2 p,  p
(3, 2) dx
P(2 p, p 2 ) y  p2  p  x  2 p 
x
y  p 2  px  2 p 2
y  px  p 2
however tangent passes through (3,2)
2  3 p  p2
p2  3 p  2  0

39. (2) Tangents from an external point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
dy
when x  2 p,  p
(3, 2) dx
P(2 p, p 2 ) y  p2  p  x  2 p 
x
y  p 2  px  2 p 2
y  px  p 2
however tangent passes through (3,2)
2  3 p  p2
p2  3 p  2  0
 p  2  p  1  0

40. (2) Tangents from an external point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
dy
when x  2 p,  p
(3, 2) dx
P(2 p, p 2 ) y  p2  p  x  2 p 
x
y  p 2  px  2 p 2
y  px  p 2
however tangent passes through (3,2)
2  3 p  p2
p2  3 p  2  0
 p  2  p  1  0
p  1 or p  2

41. (2) Tangents from an external point x2
x2  4 y y
y 4
Q(2q, q 2 ) dy x

dx 2
dy
when x  2 p,  p
(3, 2) dx
P(2 p, p 2 ) y  p2  p  x  2 p 
x
y  p 2  px  2 p 2
y  px  p 2
however tangent passes through (3,2)
2  3 p  p2
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 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 2
Q(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 2
Q(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 2
Q(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 2
Q(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 2
Q(2aq, aq 2 ) P(2ap, ap 2 ) 3 Solve simultaneously
x px  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 2
Q(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 2
Q(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 
 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 2
Q(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 
 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 2
Q(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 
 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 2
Q(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 
 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
y x 2  4ay
x

60. (4) Normal
y x 2  4ay
P(2ap, ap 2 )
x

61. (4) Normal
y x 2  4ay
P(2ap, ap 2 )
x

62. (4) Normal
y x 2  4ay
P(2ap, ap 2 ) 1 Show the slope of tangent at P is p
x

63. (4) Normal
y x 2  4ay
P(2ap, ap 2 ) 1 Show the slope of tangent at P is p
1
x 2  slope of normal is 
p

64. (4) Normal
y x 2  4ay
P(2ap, ap 2 ) 1 Show the slope of tangent at P is p
1
x 2  slope of normal is 
p
1
y  ap   x  2ap 
2
p

65. (4) Normal
y x 2  4ay
P(2ap, ap 2 ) 1 Show the slope of tangent at P is p
1
x 2  slope of normal is 
p
1
y  ap   x  2ap 
2
p
py  ap 3   x  2ap

66. (4) Normal
y x 2  4ay
P(2ap, ap 2 ) 1 Show the slope of tangent at P is p
1
x 2  slope of normal is 
p
1
y  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
y x 2  4ay
x

69. (5) Intersection of normals
y x 2  4ay
P(2ap, ap 2 )
x

70. (5) Intersection of normals
y x 2  4ay
P(2ap, ap 2 )
x

71. (5) Intersection of normals
x 2  4ay 1 Show normal at P is x  py  ap  2ap
3
y
P(2ap, ap 2 )
x

72. (5) Intersection of normals
x 2  4ay 1 Show normal at P is x  py  ap  2ap
3
y
Q(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  2ap
3
y
Q(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  2ap
3
y
2  normal at Q is x  qy  aq  2aq
3
Q(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  2ap
3
y
2  normal at Q is x  qy  aq  2aq
3
N
Q(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  2ap
3
y
2  normal at Q is x  qy  aq  2aq
3
N
Q(2aq, aq 2 ) 2 3 Solve simultaneously
P(2ap, ap )
x

77. (5) Intersection of normals
x 2  4ay 1 Show normal at P is x  py  ap  2ap
3
y
2  normal at Q is x  qy  aq  2aq
3
N
Q(2aq, aq 2 ) 2 3 Solve simultaneously
P(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  2ap
3
y
2  normal at Q is x  qy  aq  2aq
3
N
Q(2aq, aq 2 ) 2 3 Solve simultaneously
P(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  2ap
3
y
2  normal at Q is x  qy  aq  2aq
3
N
Q(2aq, aq 2 ) 2 3 Solve simultaneously
P(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  2ap
3
y
2  normal at Q is x  qy  aq  2aq
3
N
Q(2aq, aq 2 ) 2 3 Solve simultaneously
P(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 
x  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  2ap
3
y
2  normal at Q is x  qy  aq  2aq
3
N
Q(2aq, aq 2 ) 2 3 Solve simultaneously
P(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 
x  ap  p 2  pq  q 2  2   ap 3  2ap
x  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  2ap
3
y
2  normal at Q is x  qy  aq  2aq
3
N
Q(2aq, aq 2 ) 2 3 Solve simultaneously
P(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 
x  ap  p 2  pq  q 2  2   ap 3  2ap
x  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