Question
If it is given that the equation $$|z|^2-2iz+2c(a+i)=0$$ possesses solution for all $c∈R$, then the number of integral values of 'a' for which it is true is ______.
Attempt
Wrote $z=x+iy$ and substituted in the given equation to get $x=c$ as one of the condition. But after that the equation becomes $$y^2+2y+2ac+c^2=0$$. But after this I don't know what to do further.
Any hints or suggestions?
Solve for $y$ and then check for which values of $a$ the solutions are always real, i.e. for which $a$ the discriminant is always positive.