The minimum radius of the circle which is orthogonal with both the circles $x^2+y^2-12x+35=0$  and  $x^2+y^2+4x+3=0$ is

$\begin{array}{l}{x}^2+y^2-12x+35=0\\ x^2+y^2+4x+3=0\end{array}$ 

Let the circle be $x^2+y^2+2gx+2fy+c=0$  Applying condition for orthogonality
$C+12g=-35\&C-4g=-3$.    
$\therefore g=-2,C=-11,f=0,r=\sqrt{15}$