4. The center of a circle is at the left focus of the hyperbola $x^{2}-3 y^{2}=12$, and the circle passes through the origin. The coordinates of the intersection points of this circle and the hyperbola are $\qquad$
4. $(3+2 \sqrt{3}, \pm \sqrt{3+4 \sqrt{3}})$

Let $P\left(x_{1} y_{1}\right)$ be an intersection point, and the equation of the right directrix $l$ of the hyperbola is $x=3$. Draw $P M \perp l$ at $M$, then $\mid P M$ $\mid=x_{1}-3$. The eccentricity of the hyperbola is $\frac{2 \sqrt{3}}{3}\left|P F_{2}\right|=4$, where $C F_{2}$ is the right focus. Therefore, from $\frac{\left|P F_{2}\right|}{|P M|}=\frac{2 \sqrt{3}}{3}$, we can derive $x_{1}$ $=3+2 \sqrt{3}$. Thus, $y_{1}= \pm \sqrt{3+4 \sqrt{3}}$.