13.6.5 ** A chord is drawn through the focus of the parabola $y^{2}=4 x$ with an inclination angle of $\alpha$, and the length of the chord does not exceed 8. The line on which the chord lies intersects the ellipse $3 x^{2}+2 y^{2}=2$. Find the range of values for $\alpha$.


From $\left\{\begin{array}{l}y^{2}=4 x, \\ y=(x-1) \tan \alpha\end{array}\right.$, we get $x^{2} \tan ^{2} \alpha-\left(2 \tan ^{2} \alpha+4\right) x+\tan ^{2} \alpha=0$. By the definition of a parabola, the chord length $=x_{1}+1+x_{2}+1=\frac{2 \tan ^{2} \alpha+4}{\tan ^{2} \alpha}+2 \leqslant 8$, so $\tan ^{2} \alpha \geqslant 1$. From $\left\{\begin{array}{l}3 x^{2}+2 y^{2}=2, \\ y=(x-1) \tan \alpha\end{array}\right.$, we get $x^{2}\left(3+2 \tan ^{2} \alpha\right)-4 x \tan ^{2} \alpha+2 \tan ^{2} \alpha-2=0$, hence $\Delta=16 \tan ^{4} \alpha-4\left(3+2 \tan ^{2} \alpha\right)\left(2 \tan ^{2} \alpha-2\right) \geqslant 0$, which gives $\tan ^{2} \alpha \leqslant 3$. Therefore, $1 \leqslant \tan ^{2} \alpha \leqslant 3$, and thus, $\alpha \in\left[\frac{\pi}{4}, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, \frac{3 \pi}{4}\right]$.