Example 2. The parametric equation of line $L^{\prime}$ is $\left\{\begin{array}{l}x=1+t^{\prime}, \\ y=5+\sqrt{3} t^{\prime}\end{array}\right.$, it intersects the circle $x^{2}+y^{2}=16$ at points $A, B$. Find the length of the chord $AB$.
Solution 1 Using the formula $\left(L_{A B}\right) \cdot$ From Example 1, we know that the line
$$
x=1+\frac{t}{2},
$$
$L'$ in standard form is
Substituting it into
$$
x^{2}+y^{2}=16 \text { gives } t^{2}+(1+5 \sqrt{3}) t+10=0 \text {. }
$$

Therefore, $|A B|=\sqrt{[-(1+5 \sqrt{3})]^{2}-4} \times 10$
$$
=\sqrt{36+10 \sqrt{3}} \text {. }
$$

Solution 2 Using the formula $\left(L^{\prime} A_{B}\right)$. Substituting $x, y$ of $L'$ into the equation of the figure gives
$$
2 t^{\prime 2}+(1+5 \sqrt{3}) t^{\prime}+5=0 .
$$

Thus, $|A B|=\sqrt{1^{2}+(\sqrt{3})^{2}}$
$$
\begin{array}{l}
\sqrt{\left(-\frac{1+5 \sqrt{3}}{2}\right)^{2}-4 \times \frac{5}{2}} \\
=\sqrt{36+10 \sqrt{3}}
\end{array}
$$