The sum of the real parts of the complex numbers satisfying the equations   $\left|\frac{z-4i}{z-2i}\right|=1$  and   $\left|\frac{z-8+3i}{z+3i}\right|=\frac{3}{5}$ is__________

$\left|\frac{z-4i}{z-2i}\right|=1\Rightarrow z=x+3i$  using this in  
$\left|\frac{z-8+3i}{z+3i}\right|=\frac{3}{5}\Rightarrow 5|x-8+6i|=3|x+6i|$

$\Rightarrow x=8,17$
Two complex numbers  $8+3i,17+3i$
Sum of real part  $=8+17=25$