1. ## Sketching complex plane

Sketch the subset of the complex plane:

{z E C: |z - 1| = |z - i| }

I have no idea what to do. Could someone please explain?

2. Originally Posted by SyNtHeSiS
Sketch the subset of the complex plane:

{z E C: |z - 1| = |z - i| }

I have no idea what to do. Could someone please explain?
The modulus of a complex number is it's distance from the origin in the complex plane.
Pythagoras' theorem calculates the distance.

$z=x+iy$

$z-1=(x-1)+iy\Rightarrow\ |z-1|=\sqrt{(x-1)^2+y^2}=\sqrt{x^2+y^2-2x+1}$

$z-i=x+i(y-1)\Rightarrow\ |z-i|=\sqrt{x^2+(y-1)^2}=\sqrt{x^2+y^2-2y+1}$

3. Hello, SyNtHeSiS!

Did you make a sketch?

Sketch the subset of the complex plane: . $\{\,z \in C\!:\;|z - 1| \:=\:|z - i|\,\}$

Hint:

On the Argand diagram:

. . $|z-1|$ is the distance from $\,z$ to $(1,0).$

. . $|z - i|$ is the distance from $\,z$ to $(0,1).$

4. Thanks. Would this sketch be the graph y = x?

5. Yes, it is.

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### sketching subsets of the complex plane

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