(a) NO. Take $B$ a blue point, and $\gamma$ the circle of radius $1$ centred at $B$. Denote $\gamma(X,r)$ the circle of radius $r$ centred at $X$ (so $\gamma = \gamma(B,1)$). Then $\bigcup_{X \in \gamma} \gamma(X,1)$ is the full disk of radius $2$ centred at $B$, and the only blue point in it must be $B$ - but clearly there are circles of radius $1$ contained in it, not passing through $B$ - contradiction.

(b) Color all lines of the form $x=2k$ ,$k=0,\pm 1,\pm 2,...$ with blue colour