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
* major arc if
2
* semicircle if
2
2
9. z2
e.g .i Find the locus of w if w
,z 4
2
z2
w
z
zw z 2
10. z2
e.g .i Find the locus of w if w
,z 4
2
z2
w
z
zw z 2
z w 1 2
11. 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
12. 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
2
4
w 1
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
2
z
w 1
2
4
w 1
2
4
w 1
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
2
4
w 1
2
4
w 1
w 1
1
2
15. 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
2
4
w 1
2
4
w 1
w 1
1
2
1
locus is a circle, centre 1,0 and radius
2
1
2
2
i.e. x 1 y
4
16. z 1
ii Find the locus of z if w
and w is purely real
z 1
17. 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
18. 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
19. 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
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
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
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
If w is purely real then Imw 0
i.e. x 1 y x 1 y 0
xy y xy y 0
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
i.e. x 1 y x 1 y 0
xy y xy y 0
2y 0
y0
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
xy y xy y 0
2y 0
y0
locus is y 0, excluding 1,0
z 1 0, bottom of fraction 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 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
25. 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
z 1 0 or
x 2 1 i x 1 y i x 1 y y 2
i.e. arg
2
z 1
x 1 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
26. 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
z 1 0 or
x 2 1 i x 1 y i x 1 y y 2
i.e. arg
2
z 1
x 1 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
arg z 1 arg z 1 0 or
y
x
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
z 1 0 or
x 2 1 i x 1 y i x 1 y y 2
i.e. arg
2
z 1
x 1 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
arg z 1 arg z 1 0 or
y
-1
1
x
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
z 1 0 or
x 2 1 i x 1 y i x 1 y y 2
i.e. arg
2
z 1
x 1 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
arg z 1 arg z 1 0 or
y
-1
1
x
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
z 1 0 or
x 2 1 i x 1 y i x 1 y y 2
i.e. arg
2
z 1
x 1 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
arg z 1 arg z 1 0 or
y
-1
1
x
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
z 1 0 or
x 2 1 i x 1 y i x 1 y y 2
i.e. arg
2
z 1
x 1 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
arg z 1 arg z 1 0 or
y
-1
1
x
locus is y 0, excluding 1,0
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
z 1 0 or
x 2 1 i x 1 y i x 1 y y 2
i.e. arg
2
z 1
x 1 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
arg z 1 arg z 1 0 or
y
-1
1
x
locus is y 0, excluding 1,0
Note : locus is y 0, excluding 1,0 only
i.e. answer the original question
32. z
iii Find the locus of z if arg
z 4 6
33. z
iii Find the locus of z if arg
z 4 6
z
arg
z 4 6
34. z
iii Find the locus of z if arg
z 4 6
z
arg
z 4 6
35. z
iii Find the locus of z if arg
z 4 6
z
arg
z 4 6
arg z arg z 4
6
y
x
36. z
iii Find the locus of z if arg
z 4 6
z
arg
z 4 6
arg z arg z 4
6
y
4x
6
37. z
iii Find the locus of z if arg
z 4 6
z
arg
z 4 6
arg z arg z 4
6
y
4x
6
NOTE: arg z arg z-4
below axis
38. z
iii Find the locus of z if arg
z 4 6
z
arg
z 4 6
arg z arg z 4
6
y
2
4x
6
NOTE: arg z arg z-4
below axis
39. z
iii Find the locus of z if arg
z 4 6
z
arg
z 4 6
arg z arg z 4
6
y
2
r 4x
(2,y)
6
NOTE: arg z arg z-4
below axis
40. z
iii Find the locus of z if arg
z 4 6
z
arg
z 4 6
arg z arg z 4
6
y
2
r 4x
(2,y)
6
NOTE: arg z arg z-4
below axis
30
41. z
iii Find the locus of z if arg
z 4 6
y
z
tan 60
arg
2
z 4 6
arg z arg z 4
y
6
2
r 4x
(2,y)
6
NOTE: arg z arg z-4
below axis
30
42. z
iii Find the locus of z if arg
z 4 6
y
z
tan 60
arg
2
z 4 6
y 2 tan 60
arg z arg z 4
6
2 3
y
2
r 4x
(2,y)
6
NOTE: arg z arg z-4
below axis
30
43. z
iii Find the locus of z if arg
z 4 6
y
z
tan 60
arg
2
z 4 6
y 2 tan 60
arg z arg z 4
6
2 3
y
centre is 2,2 3
2
r 4x
(2,y)
6
NOTE: arg z arg z-4
below axis
30
44. z
iii Find the locus of z if arg
z 4 6
y
z
tan 60
arg
2
z 4 6
y 2 tan 60
arg z arg z 4
6
2 3
y
centre is 2,2 3
2
r 4x
(2,y)
6
NOTE: arg z arg z-4
below axis
30
r 2 2 2 2 3
2
45. z
iii Find the locus of z if arg
z 4 6
y
z
tan 60
arg
2
z 4 6
y 2 tan 60
arg z arg z 4
6
2 3
y
centre is 2,2 3
2
r 4x
(2,y)
6
NOTE: arg z arg z-4
below axis
30
r 2 2 2 2 3
2
r 2 16
r4
46. z
iii Find the locus of z if arg
z 4 6
y
z
tan 60
arg
2
z 4 6
y 2 tan 60
arg z arg z 4
6
2 3
y
r 2 2 2 2 3
2
r 2 16
r4
centre is 2,2 3
locus is the major arc of the circle
2
r 4x
(2,y)
6
NOTE: arg z arg z-4
below axis
x 2 y 2 3 16 formed by the
chord joining 0,0 and 4,0 but not
2
2
30 including these points.
47. z
iii Find the locus of z if arg
z 4 6
y
z
tan 60
arg
2
z 4 6
y 2 tan 60
arg z arg z 4
6
2 3
y
r 2 2 2 2 3
2
r 2 16
r4
centre is 2,2 3
locus is the major arc of the circle
2
r 4x
(2,y)
6
NOTE: arg z arg z-4
below axis
x 2 y 2 3 16 formed by the
chord joining 0,0 and 4,0 but not
2
2
30 including these points.
Patel: Exercise 4N; 5, 6
Cambridge: Exercise 1F; 10 to 20
HSC Geometrical Complex Numbers Questions