If $k>0,\left|z\right|=\left|w\right|=k,$  and $\alpha =\frac{z-\overline{w}}{k^2+z\overline{w}}$ , then Re $\left(\alpha \right)$  equals

  $\overline{\alpha }=\frac{\overline{z}-w}{k^2+\overline{z}w}$ 

But   $z\overline{z}=w\overline{w}=k^2,$thus,  $\overline{\alpha }=\frac{\frac{k^2}{z}-\frac{k^2}{\overline{w}}}{k^2+\frac{k^2}{z}\frac{k^2}{\overline{w}}}=\frac{\overline{w}-z}{z\overline{w}+k^2}=-\alpha$

                                $\Rightarrow$   $\alpha +\overline{\alpha }=0$  $\Rightarrow$ $\mathrm{Re}\left(\alpha \right)=0$