Sum of a Complex number Z with its Conjugate equals to zero. Conclusion?

Question

I saw an MCQ in a book that asks that sum of a complex number $Z$ with its Conjugate equals to zero if and only if Im$(Z)=0$. But my brain cannot absorb this answer. Because their sum equals to 2Re$(Z)$, therefore Re$(Z)$ must be zero. Isn't it?

Answer 1

Considering: $a=Re \ z$ and $b=Im \ z$ $$z+z^*=(a+i \ b)+(a-i \ b)=2a$$

$$2 \ a=0$$ $$a=0$$

Answer 2

I think there; in place of sum it must be difference:

Let $x=Re \ z$ and $y=Im \ z$ $$z-z^*=(x+i \ y)-(x-i \ y)=2i\ y =0$$

$$2 \ y=0$$ $$y=0$$