The line $\text{x+2y+c=0}$  intersects the circle  ${\text{x}}^2+y^2-4=0$ at two distinct points P and Q . Another line  $12x-6y=41$ intersects the circle  ${\text{x}}^{\text{2}}{\text{+y}}^{\text{2}}\text{-4x-2y+1=0}$ at two distinct points R and S. Find the value of ‘c’ for which the four points P,Q,R and S are concyclic.

$$\begin{gathered} {\text{L}}_1:x+2y+c=0\mathrm{...}\left(1\right) \\ {\text{L}}_2:x+12x-6y=41\mathrm{....}\left(2\right) \\ {S}_1:x^2+y^2-4=0 \\ {S}_2:x^2+y^2-4x-2y+1=0 \end{gathered}$$

$L_1\Rightarrow$  Radical axes of  $S_1$ and circle passing P, Q, R ,S
$L_2\Rightarrow$  Radical axes of and circle passing P, Q, R, S
$L_3\Rightarrow$  Radical axes of  $S_{1}and{S}_2$ given by
 $$\begin{gathered} {\text{S}}_{\text{1}}{\text{-S}}_{\text{2}}\text{=0 } \\ \Rightarrow \text{4x+2y-5=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{.....}\left(3\right) \end{gathered}$$ 
Radical axes of 3 circle is concurrent 
$\Rightarrow \left|\begin{array}{ccc}1& 2& c\\ 12& -6& -41\\ 4& 2& -5\end{array}\right|=0$  $\Rightarrow c=2$