Given that \( \Large y = 2x+c \) ...(i)

\( \Large x^{2}+y^{2}=16 \) ...(ii)

We Know that, if \( \Large y = mx+c \) is tangent to the circle \( \Large x^{2}+y^{2}=a^{2} \) then \( \Large c = \pm a\sqrt{1+m^{2}} \) here m = 2, a = 4

\( \Large c = \pm 4\sqrt{1+2^{2}} = \pm 4\sqrt{5} \)