If $|\mathrm{z}-3|=min\left\{\right|\mathrm{z}-1|,|\mathrm{z}-5\left|\right\}$, then Re(z) is equal to

$$\begin{gathered} \because |\mathrm{z}-3|=min\left\{\right|\mathrm{z}-1|,|\mathrm{z}-5\left|\right\} \\ \text{ Case I If }|\mathrm{z}-3|=|\mathrm{z}-1| \end{gathered}$$
On squaring both sides, we get 
$|\mathrm{z}-3{|}^2=|\mathrm{z}-1{|}^2$
$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}|\mathrm{z}{|}^2+9-2\mathrm{Re}(3\mathrm{z})\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=|\mathrm{z}{|}^2+1-2\mathrm{Re}(\mathrm{z})\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\mathrm{Re}\left(\mathrm{z}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=8\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{Re}\left(\mathrm{z}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=2\end{array}$
$\text{ Case II If }|\mathrm{z}-3|=|\mathrm{z}-5|$
On squaring both sides, we get
               $|\mathrm{z}-3{|}^2=|\mathrm{z}-5{|}^2$
$\begin{array}{l}\Rightarrow |\mathrm{z}{|}^2+9-2\mathrm{Re}(3\mathrm{z})=|\mathrm{z}{|}^2+25-2\mathrm{Re}(5\mathrm{z})\\ \Rightarrow 4\mathrm{Re}\left(\mathrm{z}\right)=16\Rightarrow \mathrm{Re}\left(\mathrm{z}\right)=4\end{array}$