Locus of mid points of chords to the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-8\mathrm{x}+6\mathrm{y}+20=0$ which are parallel to the line $3\mathrm{x}+4\mathrm{y}+5=0$ is

Locus is the diameter of the circle  perpendicular to the given line .

Equation of given line is $3\mathrm{x}+4\mathrm{y}+5=0$. The line perpendicular to given line is $$\begin{gathered} 4\mathrm{x}-3\mathrm{y}+k=0 \mathrm{Substituting} \mathrm{centre}(4,-3) \\ \Rightarrow 4\left(4\right)-3(-3)+k=0\therefore k=-25 \\ \mathrm{Required} \mathrm{locus} \mathrm{is} 4x-3y-25=0 \end{gathered}$$