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
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
27. Using Parametric Coordinates;
at Pa 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
28. Using Parametric Coordinates;
at Pa 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 bx cos ab cos 2
2
bx cos ay sin ab sin 2 ab cos 2
29. Using Parametric Coordinates;
at Pa 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 bx cos ab cos 2
2
bx cos ay sin ab sin 2 ab cos 2
x cos y sin
sin 2 cos 2
a b
30. Using Parametric Coordinates;
at Pa 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 bx cos ab cos 2
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
31. Using Parametric Coordinates;
at Pa 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 bx cos ab cos 2
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 Pa 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 bx cos ab cos by cos b sin cos
2 2 2
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
33. Using Parametric Coordinates;
at Pa 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 bx cos ab cos by cos b sin cos
2 2 2
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 ax
by
a 2 b2
a b cos sin
34. x2 y 2
For ellipse 2
2 1
a b
tangent at x1 , y1
x1 x y1 y
2
2 1
a b
35. x2 y 2
For ellipse 2
2 1
a b
tangent at x1 , y1 normal at x1 , y1
x1 x y1 y
2 1 a2 x b2 y
2 a2 b2
a b x1 y1
36. x2 y 2
For ellipse 2
2 1
a b
tangent at x1 , y1 normal at x1 , y1
x1 x y1 y
2 1 a2 x b2 y
2 a2 b2
a b x1 y1
tangent at a cos , b sin
x cos y sin
1
a b
37. x2 y 2
For ellipse 2
2 1
a b
tangent at x1 , y1 normal at x1 , y1
x1 x y1 y
2 1 a2 x b2 y
2 a2 b2
a b x1 y1
tangent at a cos , b sin normal at a cos , b sin
x cos y sin ax by
1 a 2 b2
a b cos sin
38. x2
e.g. (i) Find the equation of the tangent to the ellipse y 2 1
16
3
at the point 2,
2
39. 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
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 3 dy 2
at 2, ,
8 dx 2 dx 3
16
dy x 2
dx 16 y dy 1
dx 4 3
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
3 1
y x 2
2 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
4 3y 6 x 2
x 4 3y 8 0
43. (ii) Find the equation of the normal to the ellipse x 2cos , y sin
at the point
6
44. (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
45. (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
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
x 2 cos
dx
2 sin
d
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 y sin
dx dy
2 sin cos
d 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 dy dy d
dx dy dx d dx
2 sin cos
d d cos
2 sin
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
3
dy 2
at ,
6 dx 1
3
2
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 1 2
y x 3
at ,
6 dx 1 2 3
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 3y 3 4 x 4 3
2 4 x 2 3y 3 3 0
52. (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
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
x a cos y b sin dy b cos
dx dy
a sin b cos dx a sin
d d
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
dy
If y mx k is a tangent then m
dx
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
b cos
i.e. m
a sin
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
am sin b cos
am sin b cos 0 (1)
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)
If tangent meets ellipse at a cos , b sin then;
b sin am cos k
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
am cos b sin k (2)
59. 12 : a 2 m 2 sin 2 2abm sin cos b 2 cos 2 0 (+)
22 : a 2 m 2 cos 2 2abm sin cos b 2 sin 2 k 2
60. 12 : a 2 m 2 sin 2 2abm sin cos b 2 cos 2 0 (+)
22 : 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
61. 12 : a 2 m 2 sin 2 2abm sin cos b 2 cos 2 0 (+)
22 : 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