2. Given $M=\left\{(x, y) \mid x^{2}+2 y^{2}=3\right\}, N=\{(x, y) \mid y=m x+b\}$. If for all $m \in \mathbf{R}$, $M \bigcap N \neq \varnothing$, then the range of $b$ is
A. $\left[-\frac{\sqrt{6}}{2}, \frac{\sqrt{6}}{2}\right]$
B. $\left(-\frac{\sqrt{6}}{2}, \frac{\sqrt{6}}{2}\right)$
C. $\left(-\frac{2 \sqrt{3}}{3}, \frac{2 \sqrt{3}}{3}\right)$ D. $\left[-\frac{2 \sqrt{3}}{3}, \frac{2 \sqrt{3}}{3}\right]$
Since the line $y=m x+b$ passes through the point $(0, b)$, according to the problem, this point should be inside or on the ellipse. The ellipse $\frac{x^{2}}{3}+\frac{y^{2}}{\frac{3}{2}}=1$ intersects the $y$-axis at $\left(0, \pm \frac{\sqrt{6}}{2}\right)$, thus $b \in\left[-\frac{\sqrt{6}}{2}, \frac{\sqrt{6}}{2}\right]$. Therefore, the answer is $A$.