For example, $3 F_{1}, F_{2}$ are the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$, $a b=\sqrt{3}$, the line $l$ forms an angle $\theta$ with the $x$-axis at $F_{2}$, $\tan \theta=\frac{\sqrt{21}}{2}$, and $Q P: P F_{2}=2$. Find the equation of the hyperbola. (1991 National College Entrance Examination Question)
Let $F_{1}(-c, 0), F_{2}(c, 0)$. In the right triangle $\triangle O Q F_{2}$, from $\tan \theta=\frac{\sqrt{21}}{2}$, we get $Q\left(0,-\frac{\sqrt{21}}{2} c\right)$. Thus, $\left|P F_{1}\right|=\frac{11}{6} c,\left|P F_{2}\right|=\frac{5 c}{6},|O P|^{2}=\frac{37}{36} c^{2}$. By property 4, we have $\frac{55}{36} c^{2}-\frac{37}{36} c^{2}=b^{2}-a^{2}$, which simplifies to $b^{2}=3 a^{2}$. Combining this with the given $a^{2} b^{2}=3$, we solve to get $a^{2}=1, b^{2}=3$. Therefore, the equation of the hyperbola is $3 x^{2}-y^{2}=3$.