What is the set of points when $x$ changes in $(0,\frac{\pi}{2})$

Question

Given $z'=\dfrac{iz}{z-i}$ and $M(z')$. What is the set of points $M'$ when $x$ changes in $(0,\dfrac{\pi}{2})$ such that $z=e^{ix}$.

$$z'=\dfrac{ie^{ix}}{e^{ix}-i}=\dfrac{ie^{i(\frac{\pi}{2}+x)}}{e^{ix}-e^{i\frac{\pi}{2}}}$$

After developping i get to $$z'=\frac{1}{2sin(\frac{x+\pi}{2})} e^{i(\frac{x}{2}+\frac{3 \pi}{4})}$$

So what does $M'$ represents then?