10. Given $A, B$ are the left and right vertices of the hyperbola $E$, and point $M$ is on $E$. $\triangle A B M$ is an isosceles triangle with the vertex angle being $120^{\circ}$, then the eccentricity of $E$ is
A. $\sqrt{5}$
B. 2
C. $\sqrt{3}$
D. $\sqrt{2}$
Given that $M$ is in the first quadrant, then $M(2 a, \sqrt{3} a)$. Substituting into the hyperbola equation $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ yields $4-\frac{3 a^{2}}{b^{2}}=1 \Rightarrow \frac{a^{2}}{b^{2}}=1 \Rightarrow e=\sqrt{1+\frac{b^{2}}{a^{2}}}=\sqrt{2}$, so the answer is $D$.