If a complex number z is such that $\left(7+\mathrm{i}\right) \left(\mathrm{z}+\overline{\mathrm{z}}\right)-\left(4+\mathrm{i}\right) \left(\mathrm{z}-\overline{\mathrm{z}}\right)+116\mathrm{i}=0$ then $\mathrm{z}\overline{\mathrm{z}}=$

$\left(7+\mathrm{i}\right) \left(\mathrm{z}+\overline{\mathrm{z}}\right)-\left(4+\mathrm{i}\right) \left(\mathrm{z}-\overline{\mathrm{z}}\right)+116\mathrm{i}=0$,

$\mathrm{let} z=x+iy$

$\left(7+\mathrm{i}\right) \left(x+iy+x-iy\right)-\left(4+\mathrm{i}\right) \left(\mathrm{x}+\mathrm{iy}-\mathrm{x}+\mathrm{iy}\right)+116\mathrm{i}=0$

$\left(7+\mathrm{i}\right) 2x-\left(4+\mathrm{i}\right) \left(i2y\right)+116\mathrm{i}=0$

$\left(14x+2y\right)+i\left(2x-8y+116\right)=0$

$$\begin{gathered} 14x+2y=0\Rightarrow 28x+4y=0 \\ and 2x-8y+116\Rightarrow x-4y+58=0 \end{gathered}$$

$\Rightarrow x=-2,y=14$

$z\overline{z}=x^2+y^2=4+196=200$