Question

SHow the region on the complex plane
|z-1|>|z+2|

Answer 1

Answer:

This is a circle with radius

2

and centre

i

Explanation:

To say

|

z

−

i

|

=

2

is to say that the (Euclidean) distance between

z

and

i

is

2

.

graph{(x^2+(y-1)^2-4)(x^2+(y-1)^2-0.011) = 0 [-5.457, 5.643, -1.84, 3.71]}

Alternatively, use the definition:

|

z

|

=

√

z

¯

z

Consider

z

=

x

+

y

i

where

x

and

y

are Real.

Then

2

=

|

z

−

i

|

=

|

x

+

y

i

−

i

|

=

|

x

+

(

y

−

1

)

i

|

=

√

(

x

+

(

y

−

1

)

i

)

(

x

−

(

y

−

1

)

i

)

=

√

x

2

+

(

y

−

1

)

2

Square both ends and transpose to get:

x

2

+

(

y

−

1

)

2

=

2

2

which is the equation of a circle radius

2

with centre  

(

0

,

1

)

, i.e.

i

.