Let two parallel lines $L_1$ and $L_2$ with positive slope are tangents  to the circle $C_1:x^2+y^2-2x-16y+64=0.$  if  $L_1$  is also tangent to the circle $C_2:x^2+y^2-2x+2y-2=0.$and equation of  $L_2$ is $a\sqrt{a}x-by+c-a\sqrt{a}=0$  where $a,b,c\in N,$ then find the value of  $\frac{a+b+c}{2}$

Equation of transverse common tangent L with positive slope is  $y-5=2\sqrt{2}(x-1)$
$[\because$  slope  $=\tan \theta =\frac{4\sqrt{2}}{2}=2\sqrt{2}]$
$L=2\sqrt{2}x-y+(5-2\sqrt{2})=0$ 
Let equation of tangent L1 which is parallel to L is 
 $2\sqrt{2}x-y=C\Rightarrow \left|\frac{2\sqrt{2}\left(1\right)-8-C}{\sqrt{8}+1}\right|=1$

 $2\sqrt{2}-8-C=\pm 3$
 $\Rightarrow C=2\sqrt{2}-11,2\sqrt{2}-5$
 $\therefore equationofLis2\sqrt{2}x-y+11-2\sqrt{2}=0$