Question

Solve 2z = modulus of z + 2i

Answer 1

I assume the equation is

$2z=|z|+2i$

then let $z=x+iy$.

$2x+2iy=\sqrt{x^2+y^2}+2i$

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

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

$2x=\sqrt{x^2+1}$

$4x^2=x^2+1$

$3x^2=1$

$x=\pm\dfrac1{\sqrt3}$

but again we omit the negative solution, so that

$\boxed{z=\dfrac1{\sqrt3}+i}$