$2^{\circ}$ Let the line $\left(L^{\prime}\right)$ passing through the fixed point $P_{0}\left(x_{0}, y_{0}\right)$ intersect the conic section $f(x, y)=A x^{2}+B x y+C y^{2}$ $+D x+E y+F=0$ at points $A$ and $B$. Find the length of the chord $AB$.
Solve As shown in Figure 8, according to the relationship between $t^{\prime}$ and $t$ (•),
it is known that $\left|P_{0} A\right|=\sqrt{a^{2}+b^{2}}\left|t_{1}^{\prime}\right|$,
$$
|P \circ B|=\sqrt{a^{2}+b^{2}}\left|t_{2}^{\prime}\right| \text {, }
$$

Therefore, $|A B|=\left|P_{0} B-P_{0} A\right|=\left|\sqrt{a^{2}+b^{2}}\right| t_{2}^{\prime} \mid$
$$
\begin{aligned}
& -\sqrt{a^{2}+b^{2}}\left|t_{1}^{\prime}\right| \mid \\
= & \sqrt{a^{2}+b^{2}}\left|t_{2}^{\prime}-t_{1}^{\prime}\right| \\
= & \sqrt{a^{2}+t^{2}} \sqrt{\left(t_{1}^{\prime}+t_{2}^{\prime}\right)^{2}-4 t_{1}^{\prime} t_{2}^{\prime}}
\end{aligned}
$$

This is the chord length formula we are looking for. Here, $t_{1}^{\prime}, t_{2}^{\prime}$ are obtained by substituting the general equation $\left(L^{\prime}\right)$ into $f(x, y)=0$.

It is evident that when $\left(L^{\prime}\right)$ has $a^{2}+b^{2}=1$, then $t^{\prime}=t,\left(L^{\prime}{ }_{A B}\right)$ becomes $\left(L_{A B}\right)$, thus $\left(L^{\prime}{ }_{A B}\right)$ is universal, unifying the two forms of the parametric equation of a line, and is applicable to any positional relationship between a fixed point $P_{0}$ and a conic section, as well as different shapes of conic sections, simplifying the process of finding the chord length.