• Support PF! Buy your school textbooks, materials and every day products Here!

Complex Numbers Inequality

  • Thread starter zeromaxxx
  • Start date
  • #1
17
0

Homework Statement



Determine the region in the complex plane described by |z-2i| < |z+ i|

Homework Equations



z= x+ iy
|z|= (x2 + y2)1/2


The Attempt at a Solution



|z-2i| < |z+ i|

|z-2i|/|z+ i| < 1

|z-2i| = [(x-2i)2 + y2]1/2
|z+ i| = [(x+i)2 + y2]1/2

[(x-2i)2 + y2]1/2
--------------- < 1
[(x+i)2 + y2]1/2


[(x-2i)2 + y2]1/2*[(x+i)2 - y2]1/2
---------------------------------- < 1
[(x+i)2 + y2]1/2*[(x+i)2 - y2]1/2



Am I on the right track of solving this so far? If so how do I proceed to the next step? If not what part did I do wrong? Any feedback is appreciated!
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,705
1,722

Homework Statement



Determine the region in the complex plane described by |z-2i| < |z+ i|

Homework Equations



z= x+ iy
|z|= (x2 + y2)1/2


The Attempt at a Solution



|z-2i| < |z+ i|

|z-2i|/|z+ i| < 1

|z-2i| = [(x-2i)2 + y2]1/2
|z+ i| = [(x+i)2 + y2]1/2

[(x-2i)2 + y2]1/2
--------------- < 1
[(x+i)2 + y2]1/2


[(x-2i)2 + y2]1/2*[(x+i)2 - y2]1/2
---------------------------------- < 1
[(x+i)2 + y2]1/2*[(x+i)2 - y2]1/2



Am I on the right track of solving this so far? If so how do I proceed to the next step? If not what part did I do wrong? Any feedback is appreciated!
|z-i2|^2 is NOT (x-2i)^2 + y^2. Think about why not.

RGV
 
  • #3
dynamicsolo
Homework Helper
1,648
4
Trying to compute the inequality in Cartesian coordinates is an efficient way to make yourself crazy.

It may be more helpful to use a geometrical interpretation of the equation first in the Argand diagram. Keep in mind that | z - z0 | is the "length" of a vector from the point representing z0 to the point representing z . The equation | z - 2i | = | z + i | then describes the curve in the Argand diagram of points equidistant from 2i and -i . What is that curve like? The inequality then represents the set of points closer to 2i than to -i . Where is that region?
 

Related Threads for: Complex Numbers Inequality

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
14
Views
931
Replies
6
Views
845
Replies
4
Views
2K
  • Last Post
Replies
1
Views
701
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
3
Views
2K
Replies
5
Views
4K
Top