Let complex numbers $\alpha$ and $\frac { 1 } { \bar { \alpha } }$ lie on circles $\left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } = r ^ { 2 }$ and $\left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } = 4 r ^ { 2 }$, respectively. If $z _ { 0 } = x _ { 0 } + i y _ { 0 }$ satisfies the equation $2 \left| z _ { 0 } \right| ^ { 2 } = r ^ { 2 } + 2$, then $| \alpha | =$\\
(A) $\frac { 1 } { \sqrt { 2 } }$\\
(B) $\frac { 1 } { 2 }$\\
(C) $\frac { 1 } { \sqrt { 7 } }$\\
(D) $\frac { 1 } { 3 }$