If a circle S passing through the point (3, 4) cuts the circle $x^2 + y^2 = 36$ orthogonally then the locus of the centre of S is

$$\begin{gathered} {\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0 \mathrm{is} the required circle \mathrm{passing} \mathrm{through} (3, 4) \\ 25+6\mathrm{g}+8\mathrm{f}+\mathrm{c}=0\to \left(1\right) \\ \mathrm{and} \mathrm{cut}s o\mathrm{rthogonal}ly to the circle {\mathrm{x}}^2+{\mathrm{y}}^2=36 then \\ 2\mathrm{g}\left(0\right)+2\mathrm{f}\left(0\right)=\mathrm{c}-36 \\ \Rightarrow c=36 \\ from \left(1\right)\Rightarrow 6g+8f+25+36=0 \\ \Rightarrow -6\mathrm{g}-8\mathrm{f}-25-36=0 \\ \Rightarrow 6(-\mathrm{g})+8(-\mathrm{f})-61=0 \\ locus of centre is \\ 6\mathrm{x}+8\mathrm{y}-61=0 \end{gathered}$$