11. Given the function $y=\sin 2 x+a \cos 2 x$ whose graph is symmetric about the line $x=-\frac{\pi}{6}$, find the value of $a$.
11. According to the properties of the graph of $y=A \sin (\omega x+\varphi)$, when the function reaches its extreme values at the axis of symmetry $x=x_{0}$.

Since the extreme values of $y=\sin 2 x+a \cos 2 x$ are $\pm \sqrt{1+a^{2}}=\sin \left(2\left(-\frac{\pi}{6}\right)\right)+a \cos \left(2\left(-\frac{\pi}{6}\right)\right)$, i.e., $\pm \sqrt{1+a^{2}}=\frac{1}{2} a-\frac{\sqrt{3}}{2}$, solving for $a$ gives $a=-\frac{\sqrt{3}}{3}$.