$C_1$  and  $C_2$  are two circles whose equations are given as  $x^2+y^2=25$  and $x^2+y^2+10x+6y+1=0.$  Now $C_3$   is a variable circle which cuts $C_1$   and $C_2$  orthogonally. Tangents are drawn from the centre of  $C_3$ to $C_1,$  if the locus of the mid point of the chord of contact of tangents is $ax+3y+\frac{13}{b}(x^2+y^2)=0,$  then   $\frac{b}{a}$  is

Centre of  $C_3$  will lie on the radical axis of $C_1$  and $C_2$  which is  $10x+6y+26=0.$
Let centre of $C_3$   is  $(h,k).$ 
Equation of chord of contact through $(h,k)$  to the $C_1$  may be given as  $hx+ky=25 \mathrm{.........}\left(I\right)$
Let the mid point of the chord is $(x_1,y_1)$   the equation of the chord with the help of mid point may be given as $x{x}_1+y{y}_1=x_1^2+y_1^2$ 
Since (I) and (II) represents same straight line 
$\Rightarrow \frac{h}{25}=\frac{x_1}{x_1^2+y_1^2},\frac{k}{25}=\frac{y_1}{x_1^2+y_1^2} \mathrm{.......}\left(II\right)$ 
Since $(h,k)$  lie on the radical axis  $10\left(\frac{25x_1}{x_1^2+y_1^2}\right)+6\left(\frac{25y_1}{x_1^2+y_1^2}\right)+26=0$
the locus of $(x_1,y_1)$  is  $5x+3y+\frac{13}{25}(x^2+y^2)=0$