1. Given a hyperbola centered at the origin with its real axis on the $x$-axis, the equations of its asymptotes are $y= \pm \frac{\sqrt{6}}{3} x$. The line $y=2 x+ \frac{\sqrt{210}}{3}$ intersects the hyperbola, and the length of the chord obtained is 4. Find the equation of the hyperbola.
1. From the asymptote equations $y= \pm \frac{\sqrt{6}}{3} x$, we can set the hyperbola equation as $\frac{x^{2}}{3 m}-\frac{y^{2}}{2 m}=1$. Substituting $y=2 x+\frac{\sqrt{210}}{3}$ into the hyperbola equation, we get $5 x^{2}+2 \sqrt{210} x+35+3 m=0$. By the property $1,4=\frac{1}{5} \sqrt{\left(1+2^{2}\right)\left[(2 \sqrt{210})^{2}-20(35+3 m)\right]}$, we find $m=1$, so the required hyperbola equation is $\frac{x^{2}}{3}-\frac{y^{2}}{2}=1$.