[JEE Advanced 2013] source truncated alpha and 1 alpha lie on the circles x x 0 2

[Source truncated] $\alpha$ and $\frac1\alpha$ lie on the circles $(x-x_0)^2+(y-y_0)^2=r^2$ and $(x-x_0)^2+(y-y_0)^2=4r^2$ respectively; $z_0=x_0+iy_0$ satisfies $2|z_0|^2=r^2+2$. Find $|\alpha|$.

(a) $\frac1{\sqrt2}$
(b) $\frac12$
(c) $\frac1{\sqrt7}$
(d) $\frac13$