Consider a circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 4 x - 2 y = \alpha - 5$. Let its mirror image in the line $y = 2 x + 1$ be another circle $C _ { 2 } : 5 x ^ { 2 } + 5 y ^ { 2 } - 10 f x - 10 g y + 36 = 0$. Let $r$ be the radius of $C _ { 2 }$. Then $\alpha + r$ is equal to $\_\_\_\_$