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?

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- Oct 18th 2010, 02:30 PMSyNtHeSiSSketching 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? - Oct 18th 2010, 03:10 PMArchie Meade
The modulus of a complex number is it's distance from the origin in the complex plane.

Pythagoras' theorem calculates the distance.

$\displaystyle z=x+iy$

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

$\displaystyle z-i=x+i(y-1)\Rightarrow\ |z-i|=\sqrt{x^2+(y-1)^2}=\sqrt{x^2+y^2-2y+1}$ - Oct 18th 2010, 04:15 PMSoroban
Hello, SyNtHeSiS!

Did you make a sketch?

Quote:

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

**Hint:**

On the Argand diagram:

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

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

- Oct 19th 2010, 04:56 AMSyNtHeSiS
Thanks. Would this sketch be the graph y = x?

- Oct 19th 2010, 08:00 AMArchie Meade
Yes, it is.