2. Locus and Complex Numbers
f z , find the locus of or z
given some condition for or z
3. Locus and Complex Numbers
f z , find the locus of or z
given some condition for or z
(Make the condition the subject)
4. Locus and Complex Numbers
f z , find the locus of or z
given some condition for or z
(Make the condition the subject)
is purely real Im 0, arg 0 or
5. Locus and Complex Numbers
f z , find the locus of or z
given some condition for or z
(Make the condition the subject)
is purely real Im 0, arg 0 or
is purely imaginary Re 0, arg
2
6. Locus and Complex Numbers
f z , find the locus of or z
given some condition for or z
(Make the condition the subject)
is purely real Im 0, arg 0 or
is purely imaginary Re 0, arg
2
linear function
arg locus is an arc of a circle
linear function
7. Locus and Complex Numbers
f z , find the locus of or z
given some condition for or z
(Make the condition the subject)
is purely real Im 0, arg 0 or
is purely imaginary Re 0, arg
2
linear function
arg locus is an arc of a circle
linear function
* minor arc if
2
8. Locus and Complex Numbers
f z , find the locus of or z
given some condition for or z
(Make the condition the subject)
is purely real Im 0, arg 0 or
is purely imaginary Re 0, arg
2
linear function
arg locus is an arc of a circle
linear function
* minor arc if
2
* major arc if
2
9. Locus and Complex Numbers
f z , find the locus of or z
given some condition for or z
(Make the condition the subject)
is purely real Im 0, arg 0 or
is purely imaginary Re 0, arg
2
linear function
arg locus is an arc of a circle
linear function
* minor arc if
2
* major arc if
2
* semicircle if
2
11. z2
e.g .i Find the locus of w if w ,z 4
2
z2
w
z
12. z2
e.g .i Find the locus of w if w ,z 4
2
z2
w
z
zw z 2
13. z2
e.g .i Find the locus of w if w ,z 4
2
z2
w
z
zw z 2
z w 1 2
14. z2
e.g .i Find the locus of w if w ,z 4
2
z2
w
z
zw z 2
z w 1 2
2
z
w 1
15. z2
e.g .i Find the locus of w if w ,z 4
2
z2 2
w 4
z w 1
zw z 2
z w 1 2
2
z
w 1
16. z2
e.g .i Find the locus of w if w ,z 4
2
z2 2
w 4
z w 1
zw z 2 2
z w 1 2 4
w 1
2
z
w 1
17. z2
e.g .i Find the locus of w if w ,z 4
2
z2 2
w 4
z w 1
zw z 2 2
z w 1 2 4
w 1
2
z w 1
1
w 1 2
18. z2
e.g .i Find the locus of w if w ,z 4
2
z2 2
w 4
z w 1
zw z 2 2
z w 1 2 4
w 1
2
z w 1
1
w 1 2
1
locus is a circle, centre 1,0 and radius
2
1
i.e. x 1 y
2 2
4
19. z 1
ii Find the locus of z if w and w is purely real
z 1
20. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy
w
x 1 iy x 1 iy
21. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy
w
x 1 iy x 1 iy
x 2
1 i x 1 y i x 1 y y 2
x 12 y 2
22. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy
w
x 1 iy x 1 iy
x 2
1 i x 1 y i x 1 y y 2
x 12 y 2
If w is purely real then Imw 0
23. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy
w
x 1 iy x 1 iy
x 2
1 i x 1 y i x 1 y y 2
x 12 y 2
If w is purely real then Imw 0
i.e. x 1 y x 1 y 0
24. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy
w
x 1 iy x 1 iy
x 2
1 i x 1 y i x 1 y y 2
x 12 y 2
If w is purely real then Imw 0
i.e. x 1 y x 1 y 0
xy y xy y 0
25. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy
w
x 1 iy x 1 iy
x 2
1 i x 1 y i x 1 y y 2
x 12 y 2
If w is purely real then Imw 0
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0
26. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy
w
x 1 iy x 1 iy
x 2
1 i x 1 y i x 1 y y 2
x 12 y 2
If w is purely real then Imw 0
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0
locus is y 0, excluding 1,0
z 1 0, bottom of fraction 0
27. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy OR If w is purely real then arg w 0 or
w
x 1 iy x 1 iy
x 2
1 i x 1 y i x 1 y y 2
x 12 y 2
If w is purely real then Imw 0
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0
locus is y 0, excluding 1,0
z 1 0, bottom of fraction 0
28. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy OR If w is purely real then arg w 0 or
w
x 1 iy x 1 iy
x 2 1 i x 1 y i x 1 y y 2 z 1 0 or
i.e. arg
x 1 y 2
2 z 1
If w is purely real then Imw 0
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0
locus is y 0, excluding 1,0
z 1 0, bottom of fraction 0
29. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy OR If w is purely real then arg w 0 or
w
x 1 iy x 1 iy
x 2 1 i x 1 y i x 1 y y 2 z 1 0 or
i.e. arg
x 1 y 2
2 z 1
If w is purely real then Imw 0 arg z 1 arg z 1 0 or
y
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0 x
locus is y 0, excluding 1,0
z 1 0, bottom of fraction 0
30. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy OR If w is purely real then arg w 0 or
w
x 1 iy x 1 iy
x 2 1 i x 1 y i x 1 y y 2 z 1 0 or
i.e. arg
x 1 y 2
2 z 1
If w is purely real then Imw 0 arg z 1 arg z 1 0 or
y
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0 -1 1 x
locus is y 0, excluding 1,0
z 1 0, bottom of fraction 0
31. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy OR If w is purely real then arg w 0 or
w
x 1 iy x 1 iy
x 2 1 i x 1 y i x 1 y y 2 z 1 0 or
i.e. arg
x 1 y 2
2 z 1
If w is purely real then Imw 0 arg z 1 arg z 1 0 or
y
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0 -1 1 x
locus is y 0, excluding 1,0
z 1 0, bottom of fraction 0
32. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy OR If w is purely real then arg w 0 or
w
x 1 iy x 1 iy
x 2 1 i x 1 y i x 1 y y 2 z 1 0 or
i.e. arg
x 1 y 2
2 z 1
If w is purely real then Imw 0 arg z 1 arg z 1 0 or
y
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0 -1 1 x
locus is y 0, excluding 1,0
z 1 0, bottom of fraction 0
33. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy OR If w is purely real then arg w 0 or
w
x 1 iy x 1 iy
x 2 1 i x 1 y i x 1 y y 2 z 1 0 or
i.e. arg
x 1 y 2
2 z 1
If w is purely real then Imw 0 arg z 1 arg z 1 0 or
y
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0 -1 1 x
locus is y 0, excluding 1,0
z 1 0, bottom of fraction 0
locus is y 0, excluding 1,0
34. z 1
ii Find the locus of z if w and w is purely real
z 1
x 1 iy x 1 iy OR If w is purely real then arg w 0 or
w
x 1 iy x 1 iy
x 2 1 i x 1 y i x 1 y y 2 z 1 0 or
i.e. arg
x 1 y 2
2 z 1
If w is purely real then Imw 0 arg z 1 arg z 1 0 or
y
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0 -1 1 x
locus is y 0, excluding 1,0
z 1 0, bottom of fraction 0
locus is y 0, excluding 1,0
Note : locus is y 0, excluding 1,0 only
i.e. answer the original question
35. z
iii Find the locus of z if arg
z 4 6
36. z
iii Find the locus of z if arg
z 4 6
arg z
z 4 6
37. z
iii Find the locus of z if arg
z 4 6
arg z
z 4 6
38. z
iii Find the locus of z if arg
z 4 6
arg z
z 4 6
arg z arg z 4
6
y
x
39. z
iii Find the locus of z if arg
z 4 6
arg z
z 4 6
arg z arg z 4
6
y
4x
6
40. z
iii Find the locus of z if arg
z 4 6
arg z
z 4 6
arg z arg z 4
6
y
4x
6
NOTE: arg z arg z-4
below axis
41. z
iii Find the locus of z if arg
z 4 6
arg z
z 4 6
arg z arg z 4
6
y
2
4x
6
NOTE: arg z arg z-4
below axis
42. z
iii Find the locus of z if arg
z 4 6
arg z
z 4 6
arg z arg z 4
6
y
2
r 4x
(2,y)
6
NOTE: arg z arg z-4
below axis
43. z
iii Find the locus of z if arg
z 4 6
arg z
z 4 6
arg z arg z 4
6
y
2
r 4x
(2,y)
30
6
NOTE: arg z arg z-4
below axis
44. z
iii Find the locus of z if arg
z 4 6
arg z y
tan 60
z 4 6 2
arg z arg z 4
6
y
2
r 4x
(2,y)
30
6
NOTE: arg z arg z-4
below axis
45. z
iii Find the locus of z if arg
z 4 6
arg z y
tan 60
z 4 6 2
y 2 tan 60
arg z arg z 4
6
y 2 3
2
r 4x
(2,y)
30
6
NOTE: arg z arg z-4
below axis
46. z
iii Find the locus of z if arg
z 4 6
arg z y
tan 60
z 4 6 2
y 2 tan 60
arg z arg z 4
6
y 2 3
centre is 2,2 3
2
r 4x
(2,y)
30
6
NOTE: arg z arg z-4
below axis
47. z
iii Find the locus of z if arg
z 4 6
arg z y
tan 60 r 2 2 2 2 3
2
z 4 6 2
y 2 tan 60
arg z arg z 4
6
y 2 3
centre is 2,2 3
2
r 4x
(2,y)
30
6
NOTE: arg z arg z-4
below axis
48. z
iii Find the locus of z if arg
z 4 6
arg z y
tan 60 r 2 2 2 2 3
2
z 4 6 2
r 2 16
arg z arg z 4 y 2 tan 60
6 r4
y 2 3
centre is 2,2 3
2
r 4x
(2,y)
30
6
NOTE: arg z arg z-4
below axis
49. z
iii Find the locus of z if arg
z 4 6
arg z y
tan 60 r 2 2 2 2 3
2
z 4 6 2
r 2 16
arg z arg z 4 y 2 tan 60
6 r4
y 2 3
centre is 2,2 3
locus is the major arc of the circle
2
x 2 y 2 3 16 formed by the
2 2
r 4x
chord joining 0,0 and 4,0 but not
(2,y)
30 including these points.
6
NOTE: arg z arg z-4
below axis
50. z
iii Find the locus of z if arg
z 4 6
arg z y
tan 60 r 2 2 2 2 3
2
z 4 6 2
r 2 16
arg z arg z 4 y 2 tan 60
6 r4
y 2 3
centre is 2,2 3
locus is the major arc of the circle
2
x 2 y 2 3 16 formed by the
2 2
r 4x
chord joining 0,0 and 4,0 but not
(2,y)
30 including these points.
6 Exercise 4N; 5, 6
Exercise 4O; 3 to 10, 12, 13a, 14, 17,
NOTE: arg z arg z-4 20b, 21a, 22, 25, 26
below axis HSC Geometrical Complex Numbers Questions