2021/005) Solve $ |z - i | = |z + i |$

Method 1

Let $z = x + iy$
Then we have $ |x + iy - i | = |x + iy + i |$
Or $ |x + i(y -1) i | = |x + i(y + 1) | $
Knowing that $|x + iy| = \sqrt{x^2+y^2} $ we get
$\sqrt{x^2 + (y-1)^2} = x^2 + (y+1)^2$
Or $x^2 + (y-1)^2 = x^2 + (y+1)^2$
Or $(y-1)^2 = (y+1)^2$
or $y^2 - 2y + 1 = y^2 + 2y +1$ or $4y=0$ or $y = 0$
so the solution set is x + 0i

Another method that is Method 2 

We have z equidistant from the points (0,1), (0, -1) in the complex plane
So the point z lies on the perpendicular bisector of the segment joining (0,1) and (0,-1) .
The midpoint of the segment is (0,0) and as this is the y axis the line that is perpendicular
to this is the x axis or any point (x,0) equivalent to the number x+0i is the solution