x2 t01 12 locus & complex nos 3 (2011)
TRANSCRIPT
Locus and Complex Numbers , find the locus of or f z z
given some condition for or z
(Make the condition the subject)
Locus and Complex Numbers
or 0arg,0Imrealpurely is
, find the locus of or f z z
given some condition for or z
(Make the condition the subject)
Locus and Complex Numbers
or 0arg,0Imrealpurely is
2
arg,0Reimaginarypurely is
, find the locus of or f z z
given some condition for or z
(Make the condition the subject)
Locus and Complex Numbers
or 0arg,0Imrealpurely is
2
arg,0Reimaginarypurely is
linear functionarg locus is an arc of a circle
linear function
, find the locus of or f z z
given some condition for or z
(Make the condition the subject)
Locus and Complex Numbers
or 0arg,0Imrealpurely is
2
arg,0Reimaginarypurely is
linear functionarg locus is an arc of a circle
linear function
2 if arcminor *
, find the locus of or f z z
given some condition for or z
(Make the condition the subject)
Locus and Complex Numbers
or 0arg,0Imrealpurely is
2
arg,0Reimaginarypurely is
linear functionarg locus is an arc of a circle
linear function
2 if arcminor *
2 if arcmajor *
, find the locus of or f z z
given some condition for or z
(Make the condition the subject)
Locus and Complex Numbers
or 0arg,0Imrealpurely is
2
arg,0Reimaginarypurely is
linear functionarg locus is an arc of a circle
linear function
2 if arcminor *
2 if arcmajor *
2 if semicircle *
, find the locus of or f z z
given some condition for or z
(Make the condition the subject)
z
zw
2
4
11 i.e.
2
1 radius and 1,0 centre circle, a is locus
22
yx
4,2
2 if of locus theFind ..
z
zwwige
2 zzw
21 wz
1
2
wz
4
1
2
w
41
2
w
2
11 w
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
0
02
y
y
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
0
02
y
y
locus is 0, excluding 1,0y
1 0,bottom of fraction 0 z
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
0
02
y
y
locus is 0, excluding 1,0y
1 0,bottom of fraction 0 z
OR or 0arg then realpurely is If ww
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
0
02
y
y
locus is 0, excluding 1,0y
1 0,bottom of fraction 0 z
OR or 0arg then realpurely is If ww
or 01
1arg..
z
zei
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
0
02
y
y
locus is 0, excluding 1,0y
1 0,bottom of fraction 0 z
OR or 0arg then realpurely is If ww
or 01
1arg..
z
zei
or 01arg1arg zzy
x
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
0
02
y
y
locus is 0, excluding 1,0y
1 0,bottom of fraction 0 z
OR or 0arg then realpurely is If ww
or 01
1arg..
z
zei
or 01arg1arg zzy
x 1 -1
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
0
02
y
y
locus is 0, excluding 1,0y
1 0,bottom of fraction 0 z
OR or 0arg then realpurely is If ww
or 01
1arg..
z
zei
or 01arg1arg zzy
x 1 -1
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
0
02
y
y
locus is 0, excluding 1,0y
1 0,bottom of fraction 0 z
OR or 0arg then realpurely is If ww
or 01
1arg..
z
zei
or 01arg1arg zzy
x 1 -1
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
0
02
y
y
locus is 0, excluding 1,0y
1 0,bottom of fraction 0 z
OR or 0arg then realpurely is If ww
or 01
1arg..
z
zei
or 01arg1arg zzy
x 1 -1
locus is 0, excluding 1,0y
iyx
iyx
iyx
iyxw
1
1
1
1
realpurely is and 1
1 if of locus theFind w
z
zwzii
22
22
1
111
yx
yyxiyxix
0Im then realpurely is If ww
011.. yxyxei
0 yxyyxy
0
02
y
y
locus is 0, excluding 1,0y
1 0,bottom of fraction 0 z
OR or 0arg then realpurely is If ww
or 01
1arg..
z
zei
or 01arg1arg zzy
x 1 -1
locus is 0, excluding 1,0y : locus is 0, excluding 1,0 only
i.e. answer the original question
Note y
64
arg if of locus theFind
z
zziii
64
arg
z
z
6
4argarg
zz
y
x 4
6
axis below
4argarg
z-zNOTE:
2
(2,y)
r
64
arg if of locus theFind
z
zziii
64
arg
z
z
6
4argarg
zz
y
x 4
6
axis below
4argarg
z-zNOTE:
2
(2,y)
r
30
64
arg if of locus theFind
z
zziii
64
arg
z
z
6
4argarg
zz
y
x 4
6
axis below
4argarg
z-zNOTE:
2
(2,y)
r
30
60tan2
y
64
arg if of locus theFind
z
zziii
64
arg
z
z
6
4argarg
zz
y
x 4
6
axis below
4argarg
z-zNOTE:
2
(2,y)
r
30
60tan2
y
32
60tan2
y
64
arg if of locus theFind
z
zziii
64
arg
z
z
6
4argarg
zz
y
x 4
6
axis below
4argarg
z-zNOTE:
2
(2,y)
r
30
60tan2
y
32
60tan2
y
32,2 is centre
64
arg if of locus theFind
z
zziii
64
arg
z
z
6
4argarg
zz
y
x 4
6
axis below
4argarg
z-zNOTE:
2
(2,y)
r
30
60tan2
y
32
60tan2
y
32,2 is centre
222 322 r
64
arg if of locus theFind
z
zziii
64
arg
z
z
6
4argarg
zz
y
x 4
6
axis below
4argarg
z-zNOTE:
2
(2,y)
r
30
60tan2
y
32
60tan2
y
32,2 is centre
222 322 r
4
162
r
r
64
arg if of locus theFind
z
zziii
64
arg
z
z
6
4argarg
zz
y
x 4
6
axis below
4argarg
z-zNOTE:
2
(2,y)
r
30
60tan2
y
32
60tan2
y
32,2 is centre
222 322 r
4
162
r
r
points. theseincluding
notbut 4,0 and 0,0 joining chord
by the formed 16322
circle theof arcmajor theis locus
22
yx
64
arg if of locus theFind
z
zziii
64
arg
z
z
6
4argarg
zz
y
x 4
6
axis below
4argarg
z-zNOTE:
2
(2,y)
r
30
60tan2
y
32
60tan2
y
32,2 is centre
222 322 r
4
162
r
r
points. theseincluding
notbut 4,0 and 0,0 joining chord
by the formed 16322
circle theof arcmajor theis locus
22
yx
Exercise 4N; 5, 6
Exercise 4O; 3 to 10, 12, 13a, 14a
HSC Geometrical Complex Numbers Questions