Complex Numbers Inequality: Solving |z-2i| < |z+ i| in the Argand Diagram

[zeromaxxx]

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!

 

[Ray Vickson]

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

 

[dynamicsolo]

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?