Example 4. Find the length of the chord obtained by the intersection of the line $\left\{\begin{array}{l}x=-3+2 t, \\ y=3 t\end{array}\right.$ and the ellipse $2 x^{2}+4 x y+5 y^{2}-4 x-22 y+7=0$.
Solve the system of equations by substituting the line equations into the circle equation:
$$
\left\{\begin{array}{l}
x=-3+2 t, \text { into } \\
y=3 t
\end{array}\right.
$$
we get
$$
\begin{aligned}
77 t^{2}-134 t & +37=0 . \\
\left|t_{2}-t_{1}\right| & =\sqrt{\left(\frac{134}{77}\right)^{2}-\frac{4 \times 37}{77}} \\
& =\sqrt{\frac{660}{5029}}=\frac{4}{77} \sqrt{10} .
\end{aligned}
$$

The chord length \( d \) is given by
$$
d=\frac{4}{77} \sqrt{410} \cdot \sqrt{2^{2}+3^{2}}
$$
$$
=\frac{4}{77} \sqrt{5330} \text {. }
$$