If z is a complex number which simultaneously satisfies the equations $3|\mathrm{z}-12|=5|\mathrm{z}-8\mathrm{i}|\text{ and }|\mathrm{z}-4|=|\mathrm{z}-8|$, where $\mathrm{i}=\sqrt{-1}$then Im (z) can be

$\begin{array}{l}\because 3|\mathrm{z}-12|=5|\mathrm{z}-8\mathrm{i}|\\ \therefore 9|\mathrm{z}-12{|}^2=25|\mathrm{z}-8\mathrm{i}{|}^2\\ \Rightarrow 9(\mathrm{z}-12)(\overline{\mathrm{z}}-12)=25(\mathrm{z}-8\mathrm{i})(\overline{\mathrm{z}}+8\mathrm{i})\\ \Rightarrow 9(\mathrm{z}\overline{\mathrm{z}}-12(\mathrm{z}+\overline{\mathrm{z}})+144)=25(\mathrm{z}\overline{\mathrm{z}}+8\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}})+64)\\ \Rightarrow 16\mathrm{z}\overline{\mathrm{z}}+108(\mathrm{z}+\overline{\mathrm{z}})+200(\mathrm{z}-\overline{\mathrm{z}})\mathrm{i}+304=0\\ \Rightarrow 16\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)+216\mathrm{x}-400\mathrm{y}+304=0\end{array}$
$$\begin{gathered} \begin{array}{r}\Rightarrow 2\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)+27\mathrm{x}-50\mathrm{y}+38=0 ....\left(\mathrm{i}\right)\\ \text{ and }|\mathrm{z}-4|=|\mathrm{z}-8|\Rightarrow |\mathrm{z}-4{|}^2=|\mathrm{z}-8{|}^2\end{array} \\ \begin{array}{l}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\Rightarrow |\mathrm{z}{|}^2+16-2\mathrm{Re}(4\mathrm{z})=|\mathrm{z}{|}^2+64-2\mathrm{Re}\left(8\mathrm{z}\right)\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}8\mathrm{Re}\left(\mathrm{z}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=48\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{Re}\left(\mathrm{z}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=6\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=6 \left(\mathrm{ii}\right)\end{array} \end{gathered}$$
From Eqs. (i) and (ii) , we get 
$\begin{array}{l}2\left(36+{\mathrm{y}}^2\right)+162-50\mathrm{y}+38=0\\ \Rightarrow {\mathrm{y}}^2-25\mathrm{y}+136=0\\ \Rightarrow (\mathrm{y}-17)(\mathrm{y}-8)=0\\ \Rightarrow \mathrm{y}=17,8\\ \therefore \mathrm{Im}\left(\mathrm{z}\right)=17,8\end{array}$