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.
$\displaystyle z=x+iy$
$\displaystyle z-1=(x-1)+iy\Rightarrow\ |z-1|=\sqrt{(x-1)^2+y^2}=\sqrt{x^2+y^2-2x+1}$
$\displaystyle z-i=x+i(y-1)\Rightarrow\ |z-i|=\sqrt{x^2+(y-1)^2}=\sqrt{x^2+y^2-2y+1}$
Hello, SyNtHeSiS!
Did you make a sketch?
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 $\displaystyle (0,1).$