Question

Solve
2z = modulus of z + 2i

Answer 1

I assume the equation is

`2z=|z|+2i`

then let
`z=x+iy`.

`2x+2iy=√(x^2+y^2)+2i`

`2x+2i(y-1)=√(x^2+y^2)`

The right side is real-valued and positive, and only on the left side do we have an imaginary term, which tells us

`2(y-1)=0\implies y=1`

Then

`2x=√(x^2+1)`

`4x^2=x^2+1`

`3x^2=1`

`x=\pm\frac1{\sqrt3}`

but again we omit the negative solution, so that

`\boxed{z=\frac1{\sqrt3}+i}`