3. Let the complex numbers $z_{1}$ and $z_{2}$ correspond to points $A$ and $B$ on the complex plane, respectively, and $\left|z_{1}\right|=4, 4 z_{1}^{2}-2 z_{1} z_{2}+z_{2}^{2}=0$, with $O$ being the origin. Then the area of $\triangle O A B$ is ( ).
(A) $8 \sqrt{3}$
(B) $4 \sqrt{3}$
(C) $6 \sqrt{3}$
(D) $12 \sqrt{3}$
3. A.

Notice that, $\frac{2 z_{1}}{z_{2}}=\cos \frac{\pi}{3} \pm \mathrm{i} \sin \frac{\pi}{3}$. Then $\left|z_{2}\right|=8$, and the angle between $z_{1}$ and $z_{2}$ is $60^{\circ}$. Therefore, $S=\frac{1}{2} \times 4 \times 8 \times \frac{\sqrt{3}}{2}=8 \sqrt{3}$.