15. Given the family of curves $2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+\cos \theta+1) y=0, \theta$ being the parameter. Find the maximum value of the length of the chord intercepted by the line $y=2 x$ on this family of curves.
15. Obviously, $y=2x$ intersects with a member of the curve family at the origin, and the x-coordinate of the other intersection point is $x=\frac{8 \sin \theta + \cos \theta + 1}{2 \sin \theta - \cos \theta + 3}$. Using the universal formula, let $\tan \frac{\theta}{2} = t$, then $x = \frac{8t + 1}{2t^2 + 2t + 1}$. Rearranging and using the "discriminant" method, $\Delta = 4(-x^2 - 6x + 16) \geq 0$, solving this gives $-8 \leq x \leq 2$, hence $|x|_{\max} = 8$. Furthermore, by the chord length formula, the maximum chord length is $8 \sqrt{5}$.