Example 2. Find the length of the chord obtained by the intersection of the line $\left\{\begin{array}{l}x=-3+t, \\ y=-3+2 t\end{array}\right.$ and the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$.
Solve by substituting $\left\{\begin{array}{l}x=-3+t, \\ y=-3+2 t\end{array}\right.$ into the ellipse equation
$$
\begin{array}{l}
x^{2}+4 y^{2}=16 \text { to get } \\
17 t^{2}-54 t+29=0 . \\
\left|t_{2}-t_{1}\right|=\sqrt{\left(t_{1}+t_{2}\right)^{2}-4 t_{1} t_{2}} \\
=\sqrt{\left(\frac{54}{17}\right)^{2}-\frac{4 \times 29}{17}}=\frac{4}{17} \sqrt{59} . \\
\because a=1 \quad b=2, \\
\therefore \sqrt{a^{2}+b^{2}}=\sqrt{5} . \\
\text { Chord length } d=\left|t_{2}-t_{1}\right| \sqrt{a^{2}+b^{2}}=\frac{4}{17} \sqrt{295} .
\end{array}
$$