Example 4 Given the family of curves
$$
2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+\cos \theta+1) y=0
$$
( $\theta$ is a parameter). Find the maximum value of the length of the chord intercepted by the line $y=2 x$ on this family of curves.
(1995, National High School Mathematics Competition)
Solution: Given that the family of curves always passes through the origin, and the line $y=2x$ also passes through the origin, we know that the length of the chord intercepted by the family of curves on $y=2x$ depends only on the coordinates of the other intersection point of the family of curves with $y=2x$. Substituting $y=2x$ into the equation of the family of curves, we get
$$
\begin{array}{l}
2(2 \sin \theta-\cos \theta+3) x^{2}- \\
2(8 \sin \theta+\cos \theta+1) x=0 .
\end{array}
$$

When $x \neq 0$, we have
$$
\begin{array}{l}
(2 \sin \theta-\cos \theta+3) x-(8 \sin \theta+\cos \theta+1)=0 . \\
\text { Then }(2 x-8) \sin \theta-(x+1) \cos \theta+3 x-1=0 \text {. } \\
\text { Let } u=\cos \theta, v=\sin \theta \text {, we get } \\
u^{2}+v^{2}=1 \text {. }
\end{array}
$$

Thus, the problem is transformed into the system of lines
$$
(2 x-8) v-(x+1) u+3 x-1=0
$$

intersecting the unit circle.
Therefore, the distance from the center of the circle to the line is no greater than the radius, i.e.,
$$
\frac{|3 x-1|}{\sqrt{(2 x-8)^{2}+(x+1)^{2}}} \leqslant 1 \text {. }
$$

Solving this, we get $-8 \leqslant x \leqslant 2$.
Substituting $x=-8$ into $y=2x$ gives $y=-16$.
Thus, $|x|_{\text {max }}=8,|y|_{\text {max }}=16$.
Therefore, the maximum length of the chord is $\sqrt{8^{2}+16^{2}}=8 \sqrt{5}$.
Comment: The equation $A_{1} x+B_{1} y+C_{1}+\lambda\left(A_{2} x+\right.$ $\left.B_{2} y+C_{2}\right)=0(\lambda \neq 0)$ represents the following:
(1) When $A_{1} x+B_{1} y+C_{1}=0$ and $A_{2} x+B_{2} y+$ $C_{2}=0$ intersect, it represents a line passing through the intersection point of the two lines;
(2) When $A_{1} x+B_{1} y+C_{1}=0$ and $A_{2} x+B_{2} y+$ $C_{2}=0$ are parallel, it represents a line parallel to the two lines.