A circle S passes through the point (0,1) and is orthogonal to circles $(\mathrm{x}-1{)}^2+{\mathrm{y}}^2=16\text{ and }{\mathrm{x}}^2+{\mathrm{y}}^2=1$ then 

$\text{ S= }{\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0 \mathrm{be} \mathrm{the} \mathrm{required} \mathrm{circle}$
S cuts the circle the given circles $x^2+{\mathrm{y}}^2-2\mathrm{x}-15=0\text{ and }{\mathrm{x}}^2+{\mathrm{y}}^2-1=0$
orthogonally $$\begin{gathered} 2g(-1)+2f\left(0\right)=c+-15 \\ 2g\left(0\right)+2f\left(0\right)=c-1 \\ \Rightarrow \mathrm{c}=1,\mathrm{g}=7 \end{gathered}$$

S passes through  $(0,1)\Rightarrow \mathrm{f}=-1$
$\mathrm{S}\text{ is }{\mathrm{x}}^2+{\mathrm{y}}^2+14\mathrm{x}-2\mathrm{y}+1=0$