函数 $f(x)=a \sin x+b \cos x,(a b \neq 0)$ 的图像关于 $x=\frac{\pi}{6}$ 对称, 且 $f\left(x_0\right)=\frac{8}{5} a$, 则
$\text{A.}$ $b=\sqrt{3} a$
$\text{B.}$ $\cos \left(\frac{\pi}{6}-x_0\right)=\frac{4}{5}$
$\text{C.}$ $\cos \left(\frac{\pi}{3}-2 x_0\right)=\frac{24}{25}$
$\text{D.}$ $\sin \left(2 x_0+\frac{\pi}{6}\right)=\frac{7}{25}$