x2 t01 12 locus & complex nos 3 (2010)

Locus and Complex Numbers

Locus and Complex Numbers , find the locus of or f z z

given some condition for or z

Locus and Complex Numbers , find the locus of or f z z


(Make the condition the subject)


or 0arg,0Imrealpurely is

, find the locus of or f z z





2

arg,0Reimaginarypurely is






2


linear functionarg locus is an arc of a circle

linear function






2



linear function

2 if arcminor *






2



linear function

2 if arcminor *

2 if arcmajor *






2



linear function

2 if arcminor *

2 if arcmajor *

2 if semicircle *




4,2

2 if of locus theFind ..

z

zwwige

z

zw

2

4,2


z

zwwige

z

zw

2

4,2


z

zwwige

2 zzw

z

zw

2

4,2


z

zwwige

2 zzw

21 wz

z

zw

2

4,2


z

zwwige

2 zzw

21 wz

1

2

wz

z

zw

2

4,2


z

zwwige

2 zzw

21 wz

1

2

wz

4

1

2

w

z

zw

2

4,2


z

zwwige

2 zzw

21 wz

1

2

wz

4

1

2

w

41

2

w

z

zw

2

4,2


z

zwwige

2 zzw

21 wz

1

2

wz

4

1

2

w

41

2

w

2

11 w

z

zw

2

4

11 i.e.

2

1 radius and 1,0 centre circle, a is locus

22

yx

4,2


z

zwwige

2 zzw

21 wz

1

2

wz

4

1

2

w

41

2

w

2

11 w

realpurely is and 1

1 if of locus theFind w

z

zwzii

iyx

iyx

iyx

iyxw

1

1

1

1

realpurely is and 1


z

zwzii

iyx

iyx

iyx

iyxw

1

1

1

1

realpurely is and 1


z

zwzii

22

22

1

111

yx

yyxiyxix

iyx

iyx

iyx

iyxw

1

1

1

1

realpurely is and 1


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


z

zwzii

22

22

1

111

yx

yyxiyxix


011.. yxyxei

iyx

iyx

iyx

iyxw

1

1

1

1

realpurely is and 1


z

zwzii

22

22

1

111

yx

yyxiyxix


011.. yxyxei

0 yxyyxy

iyx

iyx

iyx

iyxw

1

1

1

1

realpurely is and 1


z

zwzii

22

22

1

111

yx

yyxiyxix


011.. yxyxei

0 yxyyxy

0

02

y

y

iyx

iyx

iyx

iyxw

1

1

1

1

realpurely is and 1


z

zwzii

22

22

1

111

yx

yyxiyxix


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


z

zwzii

22

22

1

111

yx

yyxiyxix


011.. yxyxei

0 yxyyxy

0

02

y

y



OR or 0arg then realpurely is If ww

iyx

iyx

iyx

iyxw

1

1

1

1

realpurely is and 1


z

zwzii

22

22

1

111

yx

yyxiyxix


011.. yxyxei

0 yxyyxy

0

02

y

y




or 01

1arg..

z

zei

iyx

iyx

iyx

iyxw

1

1

1

1

realpurely is and 1


z

zwzii

22

22

1

111

yx

yyxiyxix


011.. yxyxei

0 yxyyxy

0

02

y

y




or 01

1arg..

z

zei

or 01arg1arg zzy

x

iyx

iyx

iyx

iyxw

1

1

1

1

realpurely is and 1


z

zwzii

22

22

1

111

yx

yyxiyxix


011.. yxyxei

0 yxyyxy

0

02

y

y




or 01

1arg..

z

zei

or 01arg1arg zzy

x 1 -1

iyx

iyx

iyx

iyxw

1

1

1

1

realpurely is and 1


z

zwzii

22

22

1

111

yx

yyxiyxix


011.. yxyxei

0 yxyyxy

0

02

y

y




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


z

zziii

64

arg

z

z

64


z

zziii

64

arg

z

z

6

4argarg

zz

y

x

64


z

zziii

64

arg

z

z

6

4argarg

zz

y

x 4

6

64


z

zziii

64

arg

z

z

6

4argarg

zz

y

x 4

6

axis below

4argarg

z-zNOTE:

64


z

zziii

64

arg

z

z

6

4argarg

zz

y

x 4

6

axis below

4argarg

z-zNOTE:

2

64


z

zziii

64

arg

z

z

6

4argarg

zz

y

x 4

6

axis below

4argarg

z-zNOTE:

2

(2,y)

r

64


z

zziii

64

arg

z

z

6

4argarg

zz

y

x 4

6

axis below

4argarg

z-zNOTE:

2

(2,y)

r

30

64


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


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


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


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


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


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


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

x2 t01 12 locus & complex nos 3 (2010)

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