More Related Content More from Nigel Simmons (20) 2. Locus
Locus
The collection of all points whose location is determined by some
stated law.
3. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
4. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
y
x
5. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
y
4 x
6. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
y
4
4 x
7. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
y
4
β4 4 x
8. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
y
4
β4 4 x
β4
9. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
y
4
β4 4 x
β4
10. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
y
4
β4 4 x
β4
11. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
y
P ο¨ x, y ο©
4
β4 4 x
β4
12. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
y
P ο¨ x, y ο©
4
ο¨ x ο 0ο©2 ο« ο¨ y ο 0ο©2 ο½ 4
β4 4 x
β4
13. Locus
Locus
The collection of all points whose location is determined by some
stated law.
e.g. (i) Find the locus of the point which is always 4 units from the
origin
y
P ο¨ x, y ο©
4
ο¨ x ο 0ο©2 ο« ο¨ y ο 0ο©2 ο½ 4
β4 4 x x 2 ο« y 2 ο½ 16
β4
14. (ii) A point moves so that it is always 5 units away from the y axis
15. (ii) A point moves so that it is always 5 units away from the y axis
y
x
16. (ii) A point moves so that it is always 5 units away from the y axis
y
P ο¨ x, y ο©
x
17. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
x
18. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x
19. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
20. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
21. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
22. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y
x
23. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y y ο½ x ο«1
1
β1 x
24. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y y ο½ x ο«1
P ο¨ x, y ο©
1
β1 x
25. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y y ο½ x ο«1
P ο¨ x, y ο©
1
β1 x
26. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y y ο½ x ο«1 x ο y ο«1
P ο¨ x, y ο© ο½3
12 ο« ο¨ ο1ο©
2
1
β1 x
27. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y y ο½ x ο«1 x ο y ο«1
P ο¨ x, y ο© ο½3
12 ο« ο¨ ο1ο©
2
1
x ο y ο«1 ο½ 3 2
β1 x
28. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y y ο½ x ο«1 x ο y ο«1
P ο¨ x, y ο© ο½3
12 ο« ο¨ ο1ο©
2
1
x ο y ο«1 ο½ 3 2
β1 x
x ο y ο«1 ο½ 3 2
29. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y y ο½ x ο«1 x ο y ο«1
P ο¨ x, y ο© ο½3
12 ο« ο¨ ο1ο©
2
1
x ο y ο«1 ο½ 3 2
β1 x
x ο y ο«1 ο½ 3 2 or ο ο¨ x ο y ο« 1ο© ο½ 3 2
30. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y y ο½ x ο«1 x ο y ο«1
P ο¨ x, y ο© ο½3
12 ο« ο¨ ο1ο©
2
1
x ο y ο«1 ο½ 3 2
β1 x
x ο y ο«1 ο½ 3 2 or ο ο¨ x ο y ο« 1ο© ο½ 3 2
x ο y ο«1ο 3 2 ο½ 0
31. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y y ο½ x ο«1 x ο y ο«1
P ο¨ x, y ο© ο½3
12 ο« ο¨ ο1ο©
2
1
x ο y ο«1 ο½ 3 2
β1 x
x ο y ο«1 ο½ 3 2 or ο ο¨ x ο y ο« 1ο© ο½ 3 2
x ο y ο«1ο 3 2 ο½ 0 οx ο« y ο1 ο½ 3 2
32. (ii) A point moves so that it is always 5 units away from the y axis
y
ο¨ 0, y ο© 5 Pο¨ x, y ο©
ο¨ x ο 0ο© ο« ο¨ y ο y ο© ο½ 5
2 2
x x 2 ο½ 25
x ο½ ο±5
(iii) A point moves so that it is always 3 units away from the line y = x + 1
y y ο½ x ο«1 x ο y ο«1
P ο¨ x, y ο© ο½3
12 ο« ο¨ ο1ο©
2
1
x ο y ο«1 ο½ 3 2
β1 x
x ο y ο«1 ο½ 3 2 or ο ο¨ x ο y ο« 1ο© ο½ 3 2
x ο y ο«1ο 3 2 ο½ 0 οx ο« y ο1 ο½ 3 2
x ο y ο«1ο« 3 2 ο½ 0
33. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
34. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
y
x
35. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y
x
36. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y
ο¨ x,0 ο© x
37. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y
ο¨ 0, y ο©
ο¨ x,0 ο© x
38. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
5d
ο¨ x,0 ο© x
39. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
ο¨ x,0 ο© x
40. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x
41. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x y ο½ ο±5 x
42. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x y ο½ ο±5 x
(v) A point moves so that its distance from (1,2) is the same as its
distance from (5,β4).
43. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x y ο½ ο±5 x
(v) A point moves so that its distance from (1,2) is the same as its
distance from (5,β4).
y
x
44. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x y ο½ ο±5 x
(v) A point moves so that its distance from (1,2) is the same as its
distance from (5,β4).
y
ο¨1, 2 ο©
x
ο¨ 5, ο4 ο©
45. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x y ο½ ο±5 x
(v) A point moves so that its distance from (1,2) is the same as its
distance from (5,β4).
y
ο¨1, 2 ο©
P ο¨ x, y ο© x
ο¨ 5, ο4 ο©
46. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x y ο½ ο±5 x
(v) A point moves so that its distance from (1,2) is the same as its
distance from (5,β4).
y
ο¨1, 2 ο©
P ο¨ x, y ο© x
ο¨ 5, ο4 ο©
47. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x y ο½ ο±5 x
(v) A point moves so that its distance from (1,2) is the same as its
distance from (5,β4).
y Perpendicular bisector of (1,2) and (5,β4)
ο¨1, 2 ο©
P ο¨ x, y ο© x
ο¨ 5, ο4 ο©
48. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x y ο½ ο±5 x
(v) A point moves so that its distance from (1,2) is the same as its
distance from (5,β4).
y Perpendicular bisector of (1,2) and (5,β4)
ο4 ο 2
ο¨1, 2 ο© mAB ο½
5 ο1
P ο¨ x, y ο© ο6
x ο½
4
ο¨ 5, ο4 ο© ο3
ο½
2
49. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x y ο½ ο±5 x
(v) A point moves so that its distance from (1,2) is the same as its
distance from (5,β4).
y Perpendicular bisector of (1,2) and (5,β4)
ο4 ο 2
ο¨1, 2 ο© mAB ο½
5 ο1
P ο¨ x, y ο© ο6
x ο½
4
ο¨ 5, ο4 ο© ο3
ο½
2 2
ο required slope ο½
3
50. (iv) A point moves so that its distance from the x axis is always 5 times
as great as its distance from the y axis.
P ο¨ x, y ο©
y d
ο¨ 0, y ο©
ο¨ x ο x ο© ο« ο¨ y ο 0ο© ο½ 5 ο¨ x ο 0ο© ο« ο¨ y ο y ο©
2 2 2 2
5d
y 2 ο½ 25 x 2
ο¨ x,0 ο© x y ο½ ο±5 x
(v) A point moves so that its distance from (1,2) is the same as its
distance from (5,β4).
y Perpendicular bisector of (1,2) and (5,β4)
ο4 ο 2
ο¨1, 2 ο© mAB ο½ ο¦ 1 ο« 5 ο4 ο« 2 οΆ
M AB ο½ ο§
5 ο1 , ο·
ο¨ 2 2 οΈ
P ο¨ x, y ο© ο6
x ο½ ο½ ο¨ 3, ο1ο©
4
ο¨ 5, ο4 ο© ο3
ο½
2 2
ο required slope ο½
3
52. 2
y ο« 1 ο½ ο¨ x ο 3ο©
3
3y ο« 3 ο½ 2x ο 6
2x ο 3y ο 9 ο½ 0
53. 2
y ο« 1 ο½ ο¨ x ο 3ο©
3
3y ο« 3 ο½ 2x ο 6
2x ο 3y ο 9 ο½ 0
OR
54. 2
y ο« 1 ο½ ο¨ x ο 3ο©
3
3y ο« 3 ο½ 2x ο 6
2x ο 3y ο 9 ο½ 0
OR
ο¨ x ο 1ο© ο« ο¨ y ο 2 ο© ο½ ο¨ x ο 5ο© ο« ο¨ y ο« 4 ο©
2 2 2 2
55. 2
y ο« 1 ο½ ο¨ x ο 3ο©
3
3y ο« 3 ο½ 2x ο 6
2x ο 3y ο 9 ο½ 0
OR
ο¨ x ο 1ο© ο« ο¨ y ο 2 ο© ο½ ο¨ x ο 5ο© ο« ο¨ y ο« 4 ο©
2 2 2 2
x 2 ο 2 x ο« 1 ο« y 2 ο 4 y ο« 4 ο½ x 2 ο 10 x ο« 25 ο« y 2 ο« 8 y ο« 16
56. 2
y ο« 1 ο½ ο¨ x ο 3ο©
3
3y ο« 3 ο½ 2x ο 6
2x ο 3y ο 9 ο½ 0
OR
ο¨ x ο 1ο© ο« ο¨ y ο 2 ο© ο½ ο¨ x ο 5ο© ο« ο¨ y ο« 4 ο©
2 2 2 2
x 2 ο 2 x ο« 1 ο« y 2 ο 4 y ο« 4 ο½ x 2 ο 10 x ο« 25 ο« y 2 ο« 8 y ο« 16
8 x ο 12 y ο 36 ο½ 0
57. 2
y ο« 1 ο½ ο¨ x ο 3ο©
3
3y ο« 3 ο½ 2x ο 6
2x ο 3y ο 9 ο½ 0
OR
ο¨ x ο 1ο© ο« ο¨ y ο 2 ο© ο½ ο¨ x ο 5ο© ο« ο¨ y ο« 4 ο©
2 2 2 2
x 2 ο 2 x ο« 1 ο« y 2 ο 4 y ο« 4 ο½ x 2 ο 10 x ο« 25 ο« y 2 ο« 8 y ο« 16
8 x ο 12 y ο 36 ο½ 0
2x ο 3 y ο 9 ο½ 0
58. 2
y ο« 1 ο½ ο¨ x ο 3ο©
3
3y ο« 3 ο½ 2x ο 6
2x ο 3y ο 9 ο½ 0
OR
ο¨ x ο 1ο© ο« ο¨ y ο 2 ο© ο½ ο¨ x ο 5ο© ο« ο¨ y ο« 4 ο©
2 2 2 2
x 2 ο 2 x ο« 1 ο« y 2 ο 4 y ο« 4 ο½ x 2 ο 10 x ο« 25 ο« y 2 ο« 8 y ο« 16
8 x ο 12 y ο 36 ο½ 0
2x ο 3 y ο 9 ο½ 0
Exercise 9A; 1aceg, 3a, 4, 6, 8, 10, 11, 13, 14
