A circle passes through the point (3,4) and cuts the circle $x^2 + y^2 = a^2$orthagonally; the locus of its centre is a straight line. If the distance of this straight line from the origin is 25, then $a^2$ =___

$$\begin{gathered} \mathrm{S}:{\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0\to \left(1\right) \\ \mathrm{Centre} \mathrm{C}=(-\mathrm{g}, -\mathrm{f}) \\ Which \mathrm{is} o\mathrm{rthogonal} \mathrm{to} the \mathrm{circle} x^2+y^2-a^2=0 then \\ 2{\mathrm{gg}}^1+2{\mathrm{ff}}^1=c+c^1 \\ \Rightarrow c={\mathrm{a}}^2 \\ \left(1\right) \mathrm{passes} \mathrm{through} (3,4) \mathrm{then} \\ 9+16-6x-8y+{\mathrm{a}}^2=0 \\ 25-6\mathrm{x}-8\mathrm{y}+{\mathrm{a}}^2=0 \\ 6\mathrm{x}+8\mathrm{y}-25-{\mathrm{a}}^2=0 \\ Given that \frac{\left|-25-{\mathrm{a}}^2\right|}{10}=25 \\ \Rightarrow {\mathrm{a}}^2=225 \end{gathered}$$