Example Ten Find the equation of the hyperbola passing through the point $(-1,3)$ and having the same asymptotes as the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{9}=1$.
Let the equation of the hyperbola be $\frac{x^{2}}{4}-\frac{y^{2}}{9}=\lambda$, substituting $(-1,3)$, we get $\lambda=-\frac{3}{4}$. Thus, the equation of the hyperbola is $-\frac{x^{2}}{3}+\frac{4 y^{2}}{27}=1$.