1. Conics & Parameters
y
1) Circle
a
-a a x
x2  y2  a2
-a

2. Conics & Parameters
y
1) Circle
a
P  x, y 
-a a x
x2  y2  a2
-a

3. Conics & Parameters
y
1) Circle a
P  x, y 
a

-a x a x
x2  y2  a2
-a
x
 cos
a

4. Conics & Parameters
y
1) Circle a
P  x, y 
a

-a x a x
x2  y2  a2
-a
x
 cos
a
x  a cos

5. Conics & Parameters y
1) Circle
a
P  x, y 
a y

-a x a x
x2  y2  a2
-a
x y
 cos  sin 
a a
x  a cos y  a sin 

6. Conics & Parameters y
1) Circle
a
P  x, y 
a y

-a x a x
x2  y2  a2
-a
x y Proof:
 cos  sin 
a a x 2  y 2  a 2 cos 2   a 2 sin 2 
x  a cos y  a sin   a 2 cos 2   sin 2  
 a2

7. 2) Ellipse y
b
P  x, y 
-a a x
-b

8. 2) Ellipse y
b
P  x, y 
-a a x
-b
x2  y2  a2

9. 2) Ellipse y
b
P  x, y 

-a x a x
-b
x2  y2  a2

10. 2) Ellipse y
b
P  x, y 

-a x a x
-b
x2  y2  a2
x
 cos
a
x  a cos

11. 2) Ellipse y
b
P  x, y 
y
-a x a x
-b
x2  y2  a2
x
 cos
a
x  a cos

12. 2) Ellipse y
b
P  x, y 
y
-a x a x
-b
x2  y2  a2
x y
 cos  sin 
a b
x  a cos y  b sin 

13. 2) Ellipse y
b
P  x, y 
x2  y 2  b2
y
-a x a x
-b
x2  y2  a2
Proof:
x y
 cos  sin 
a b x 2 y 2 a 2 cos 2  b 2 sin 2 
2
 2  2

x  a cos y  b sin  a b a b2
 cos 2   sin 2 
1

14. Equation of Tangent and Normal
y
b
P x1 , y1 
-a a x
-b
x2 y2
2
 2 1
a b

15. Equation of Tangent and Normal
y
b
P x1 , y1 
-a a x
-b
x2 y2
2
 2 1
a b
2 x 2 y dy  df df dy 
 2  0  dx  dy  dx 
 
2
a b dx

16. Equation of Tangent and Normal
y
b
P x1 , y1 
-a a x
-b
x2 y2
2
 2 1
a b
2 x 2 y dy  df df dy 
 2  0  dx  dy  dx 
 
2
a b dx
2 y dy 2x
2
  2
b dx a
dy b2 x
 2
dx a y

17. Equation of Tangent and Normal
y
b
P x1 , y1 
-a a x
-b
x2 y2
2
 2 1
a b
2 x 2 y dy  df df dy 
 2  0  dx  dy  dx 
 
2
a b dx
2 y dy 2x
2
  2 at P x1 , y1 
b dx a
dy b2 x dy b 2 x1
 2  2
dx a y dx a y1

18. y
b
P x1 , y1 
-a a x
-b
tangent:
b 2 x1
y  y1   2  x  x1 
a y1

19. y
b
P x1 , y1 
-a a x
-b
tangent:
b 2 x1
y  y1   2  x  x1 
a y1
a 2 y1 y  a 2 y12  b 2 x1 x  b 2 x12
b 2 x1 x  a 2 y1 y  b 2 x12  a 2 y12

20. y
b
P x1 , y1 
-a a x
-b
tangent:
b 2 x1
y  y1   2  x  x1 
a y1
a 2 y1 y  a 2 y12  b 2 x1 x  b 2 x12
b 2 x1 x  a 2 y1 y  b 2 x12  a 2 y12
x1 x y1 y x12 y12
2
 2  2 2
a b a b

21. y
b
P x1 , y1 
-a a x
-b
tangent:
b 2 x1
y  y1   2  x  x1 
a y1
a 2 y1 y  a 2 y12  b 2 x1 x  b 2 x12
b 2 x1 x  a 2 y1 y  b 2 x12  a 2 y12
x1 x y1 y x12 y12
2
 2
 2 2
a b a b
x1 x y1 y
2
 2
1
a b

22. y
b
P x1 , y1 
-a a x
-b
tangent: normal:
b 2 x1 a 2 y1
y  y1   2  x  x1  y  y1  2  x  x1 
a y1 b x1
a 2 y1 y  a 2 y12  b 2 x1 x  b 2 x12
b 2 x1 x  a 2 y1 y  b 2 x12  a 2 y12
x1 x y1 y x12 y12
2
 2
 2 2
a b a b
x1 x y1 y
2
 2
1
a b

23. y
b
P x1 , y1 
-a a x
-b
tangent: normal:
b 2 x1 a 2 y1
y  y1   2  x  x1  y  y1  2  x  x1 
a y1 b x1
a 2 y1 y  a 2 y12  b 2 x1 x  b 2 x12 b 2 x1 y  b 2 x1 y1  a 2 y1 x  a 2 x1 y1
b 2 x1 x  a 2 y1 y  b 2 x12  a 2 y12 a 2 y1 x  b 2 x1 y  a 2 x1 y1  b 2 x1 y1
x1 x y1 y x12 y12
2
 2
 2 2
a b a b
x1 x y1 y
2
 2
1
a b

24. y
b
P x1 , y1 
-a a x
-b
tangent: normal:
b 2 x1 a 2 y1
y  y1   2  x  x1  y  y1  2  x  x1 
a y1 b x1
a 2 y1 y  a 2 y12  b 2 x1 x  b 2 x12 b 2 x1 y  b 2 x1 y1  a 2 y1 x  a 2 x1 y1
b 2 x1 x  a 2 y1 y  b 2 x12  a 2 y12 a 2 y1 x  b 2 x1 y  a 2 x1 y1  b 2 x1 y1
x1 x y1 y x12 y12 a2 x b2 y
2
 2
 2 2   a2  b2
a b a b x1 y1
x1 x y1 y
2
 2
1
a b

25. y
b
P x1 , y1 
-a a x
-b
tangent: normal:
b 2 x1 a 2 y1
y  y1   2  x  x1  y  y1  2  x  x1 
a y1 b x1
a 2 y1 y  a 2 y12  b 2 x1 x  b 2 x12 b 2 x1 y  b 2 x1 y1  a 2 y1 x  a 2 x1 y1
b 2 x1 x  a 2 y1 y  b 2 x12  a 2 y12 a 2 y1 x  b 2 x1 y  a 2 x1 y1  b 2 x1 y1
x1 x y1 y x12 y12 a2 x b2 y
2
 2
 2 2   a2  b2
a b a b x1 y1
x1 x y1 y
 1 a 2 x b2 y
a 2
b 2
  a 2e2
x1 y1

26. Using Parametric Coordinates;
at Pa cos , b sin  

27. Using Parametric Coordinates;
at Pa cos , b sin   dy ab 2 cos
 2
dx a b sin 
b cos

a sin 

28. Using Parametric Coordinates;
at Pa cos , b sin   dy ab 2 cos
 2
dx a b sin 
b cos

a sin 
tangent:
b cos
y  b sin     x  a cos 
a sin 

29. Using Parametric Coordinates;
at Pa cos , b sin   dy ab 2 cos
 2
dx a b sin 
b cos

a sin 
tangent:
b cos
y  b sin     x  a cos 
a sin 
ay sin   ab sin 2   bx cos  ab cos 2 
bx cos  ay sin   ab sin 2   ab cos 2 

30. Using Parametric Coordinates;
at Pa cos , b sin   dy ab 2 cos
 2
dx a b sin 
b cos

a sin 
tangent:
b cos
y  b sin     x  a cos 
a sin 
ay sin   ab sin 2   bx cos  ab cos 2 
bx cos  ay sin   ab sin 2   ab cos 2 
x cos y sin 
  sin 2   cos 2 
a b

31. Using Parametric Coordinates;
at Pa cos , b sin   dy ab 2 cos
 2
dx a b sin 
b cos

a sin 
tangent:
b cos
y  b sin     x  a cos 
a sin 
ay sin   ab sin 2   bx cos  ab cos 2 
bx cos  ay sin   ab sin 2   ab cos 2 
x cos y sin 
  sin 2   cos 2 
a b
x cos y sin 
 1
a b

32. Using Parametric Coordinates;
at Pa cos , b sin   dy ab 2 cos
 2
dx a b sin 
b cos

a sin 
tangent: normal:
b cos a sin 
y  b sin     x  a cos  y  b sin    x  a cos 
a sin  b cos
ay sin   ab sin 2   bx cos  ab cos 2 
bx cos  ay sin   ab sin 2   ab cos 2 
x cos y sin 
  sin 2   cos 2 
a b
x cos y sin 
 1
a b

33. Using Parametric Coordinates;
at Pa cos , b sin   dy ab 2 cos
 2
dx a b sin 
b cos

a sin 
tangent: normal:
b cos a sin 
y  b sin     x  a cos  y  b sin    x  a cos 
a sin  b cos
ay sin   ab sin 2   bx cos  ab cos 2  by cos  b 2 sin  cos
bx cos  ay sin   ab sin 2   ab cos 2   ax sin   a 2 sin  cos
x cos y sin  ax sin   by cos
  sin 2   cos 2 
a b  a 2  b 2 sin  cos
x cos y sin 
 1
a b

34. Using Parametric Coordinates;
at Pa cos , b sin   dy ab 2 cos
 2
dx a b sin 
b cos

a sin 
tangent: normal:
b cos a sin 
y  b sin     x  a cos  y  b sin    x  a cos 
a sin  b cos
ay sin   ab sin 2   bx cos  ab cos 2  by cos  b 2 sin  cos
bx cos  ay sin   ab sin 2   ab cos 2   ax sin   a 2 sin  cos
x cos y sin  ax sin   by cos
  sin 2   cos 2 
a b  a 2  b 2 sin  cos
x cos y sin 
a

b
1 ax

by
cos  sin 
 a 2  b2  a e 2 2

35. x2 y2
For ellipse 2
 2 1
a b
tangent at  x1 , y1 
x1 x y1 y
2
 2 1
a b

36. x2 y2
For ellipse 2
 2 1
a b
tangent at  x1 , y1  normal at  x1 , y1 
a2 x b2 y
 a 2  b2   a 2e2 
x1 x y1 y
2
 2 1 
a b x1 y1

37. x2 y2
For ellipse 2
 2 1
a b
tangent at  x1 , y1  normal at  x1 , y1 
a2 x b2 y
 a 2  b2   a 2e2 
x1 x y1 y
2
 2 1 
a b x1 y1
tangent at a cos , b sin  
x cos y sin 
 1
a b

38. x2 y2
For ellipse 2
 2 1
a b
tangent at  x1 , y1  normal at  x1 , y1 
a2 x b2 y
 a 2  b2   a 2e2 
x1 x y1 y
2
 2 1 
a b x1 y1
tangent at a cos , b sin   normal at a cos , b sin  
x cos y sin 
 a 2  b2   a 2e2 
ax by
 1 
a b cos  sin 

39. x2
e.g. (i) Find the equation of the tangent to the ellipse  y 2  1
16
  3
at the point  2, 
 2 

40. x2
e.g. (i) Find the equation of the tangent to the ellipse  y 2  1
16
  3
at the point  2, 
 2 
x dy
 2y  0
8 dx
dy  x

dx 16 y

41. x2
e.g. (i) Find the equation of the tangent to the ellipse  y 2  1
16
  3
at the point  2, 
 2 
x dy
 2y  0   3  dy 2
at  2, , 
8 dx  2  dx  3
16 
dy  x  2 

dx 16 y dy 1

dx 4 3

42. x2
e.g. (i) Find the equation of the tangent to the ellipse  y 2  1
16
  3
at the point  2, 
 2 
x dy
 2y  0   3  dy 2
at  2, , 
8 dx  2  dx  3
16 
dy  x  2 

dx 16 y dy 1

dx 4 3
3 1
y   x  2
2 4 3

43. x2
e.g. (i) Find the equation of the tangent to the ellipse  y 2  1
16
  3
at the point  2, 
 2 
x dy
 2y  0   3  dy 2
at  2, , 
8 dx  2  dx  3
16 
dy  x  2 

dx 16 y dy 1

dx 4 3
3 1
y   x  2
2 4 3
4 3y  6  x  2
x  4 3y  8  0

44. (ii) Find the equation of the normal to the ellipse x  2cos , y  sin 

at the point  
6

45. (ii) Find the equation of the normal to the ellipse x  2cos , y  sin 

at the point  
6
y  sin f  x 
dy
 f  x  cos f  x 
dx

46. (ii) Find the equation of the normal to the ellipse x  2cos , y  sin 

at the point  
6
y  sin f  x  y  cos f  x 
dy dy
 f  x  cos f  x    f  x  sin f  x 
dx dx

47. (ii) Find the equation of the normal to the ellipse x  2cos , y  sin 

at the point  
6
y  sin f  x  y  cos f  x 
dy dy
 f  x  cos f  x    f  x  sin f  x 
dx dx
x  2 cos
dx
 2 sin 
d

48. (ii) Find the equation of the normal to the ellipse x  2cos , y  sin 

at the point  
6
y  sin f  x  y  cos f  x 
dy dy
 f  x  cos f  x    f  x  sin f  x 
dx dx
x  2 cos y  sin 
dx dy
 2 sin   cos
d d

49. (ii) Find the equation of the normal to the ellipse x  2cos , y  sin 

at the point  
6
y  sin f  x  y  cos f  x 
dy dy
 f  x  cos f  x    f  x  sin f  x 
dx dx
x  2 cos y  sin  dy dy d
 
dx dy dx d dx
 2 sin   cos
d d cos

 2 sin 

50. (ii) Find the equation of the normal to the ellipse x  2cos , y  sin 

at the point  
6
y  sin f  x  y  cos f  x 
dy dy
 f  x  cos f  x    f  x  sin f  x 
dx dx
x  2 cos y  sin  dy dy d
 
dx dy dx d dx
 2 sin   cos
d d cos

 2 sin 
3
 dy 2
at   , 
6 dx  1
 3

2

51. (ii) Find the equation of the normal to the ellipse x  2cos , y  sin 

at the point  
6
y  sin f  x  y  cos f  x 
dy dy
 f  x  cos f  x    f  x  sin f  x 
dx dx
x  2 cos y  sin  dy dy d
 
dx dy dx d dx
 2 sin   cos
d d cos

 2 sin 
3
 dy 2 1 2
y  x  3 
at   , 
6 dx  1 2 3
 3

2

52. (ii) Find the equation of the normal to the ellipse x  2cos , y  sin 

at the point  
6
y  sin f  x  y  cos f  x 
dy dy
 f  x  cos f  x    f  x  sin f  x 
dx dx
x  2 cos y  sin  dy dy d
 
dx dy dx d dx
 2 sin   cos
d d cos

 2 sin 
3
 dy 2 1 2
y  x  3 
at   , 
6 dx  1 2 3
 3 2 3y  3  4 x  4 3

2 4 x  2 3y  3 3  0

53. (iii) Show that if y  mx  k is a tangent to the ellipse
x2 y2
 2  1, then a 2 m 2  b 2  k 2
a2 b

54. (iii) Show that if y  mx  k is a tangent to the ellipse
x2 y2
 2  1, then a 2 m 2  b 2  k 2
a2 b
x  a cos y  b sin  dy  b cos
dx dy 
  a sin   b cos dx a sin 
d d

55. (iii) Show that if y  mx  k is a tangent to the ellipse
x2 y2
 2  1, then a 2 m 2  b 2  k 2
a2 b
x  a cos y  b sin  dy  b cos
dx dy 
  a sin   b cos dx a sin 
d d
dy
If y  mx  k is a tangent then m
dx

56. (iii) Show that if y  mx  k is a tangent to the ellipse
x2 y2
 2  1, then a 2 m 2  b 2  k 2
a2 b
x  a cos y  b sin  dy  b cos
dx dy 
  a sin   b cos dx a sin 
d d
dy
If y  mx  k is a tangent then m
dx
 b cos
i.e. m 
a sin 

57. (iii) Show that if y  mx  k is a tangent to the ellipse
x2 y2
 2  1, then a 2 m 2  b 2  k 2
a2 b
x  a cos y  b sin  dy  b cos
dx dy 
  a sin   b cos dx a sin 
d d
dy
If y  mx  k is a tangent then m
dx
 b cos
i.e. m 
a sin 
am sin   b cos
am sin   b cos  0  (1)

58. (iii) Show that if y  mx  k is a tangent to the ellipse
x2 y2
 2  1, then a 2 m 2  b 2  k 2
a2 b
x  a cos y  b sin  dy  b cos
dx dy 
  a sin   b cos dx a sin 
d d
dy
If y  mx  k is a tangent then m
dx
 b cos
i.e. m 
a sin 
am sin   b cos
am sin   b cos  0  (1)
If tangent meets ellipse at a cos , b sin   then;
b sin   am cos  k

59. (iii) Show that if y  mx  k is a tangent to the ellipse
x2 y2
 2  1, then a 2 m 2  b 2  k 2
a2 b
x  a cos y  b sin  dy  b cos
dx dy 
  a sin   b cos dx a sin 
d d
dy
If y  mx  k is a tangent then m
dx
 b cos
i.e. m 
a sin 
am sin   b cos
am sin   b cos  0  (1)
If tangent meets ellipse at a cos , b sin   then;
b sin   am cos  k
am cos  b sin    k  (2)

60. 12 : a 2 m 2 sin 2   2abm sin  cos  b 2 cos 2   0 (+)
22 : a 2 m 2 cos 2   2abm sin  cos  b 2 sin 2   k 2

61. 12 : a 2 m 2 sin 2   2abm sin  cos  b 2 cos 2   0 (+)
22 : a 2 m 2 cos 2   2abm sin  cos  b 2 sin 2   k 2
a 2 m 2 sin 2   cos 2    b 2 sin 2   cos 2    k 2

62. 12 : a 2 m 2 sin 2   2abm sin  cos  b 2 cos 2   0 (+)
22 : a 2 m 2 cos 2   2abm sin  cos  b 2 sin 2   k 2
a 2 m 2 sin 2   cos 2    b 2 sin 2   cos 2    k 2
a 2m2  b2  k 2
Exercise 6C; 1, 3, 11, 15, 18a