\((1)\)已知函数\(f\left( x \right)={{\log }_{2}}\left( {{x}^{2}}+a \right)\)，若\(f\left( 3 \right)=1\)，则\(a=\)________．
\((2)\)若\(x\)，\(y\)满足约束条件\(\begin{cases}\begin{matrix}x-2y-2\leqslant 0 \\ x-y+1\geqslant 0\end{matrix} \\ y\leqslant 0\end{cases} \)，则\(z=3x+2y\)的最大值为________．
\((3)\)直线\(y=x+1\)与圆\({{x}^{2}}+{{y}^{2}}+2y-3=0\)交于\(A\)，\(B\)两点，则\(\left| AB \right|=\)________．
\((4)\triangle ABC\)的内角\(A\)，\(B\)，\(C\)的对边分别为\(a\)，\(b\)，\(c\)，已知\(b\sin C+c\sin B=4a\sin B\sin C\)，\({{b}^{2}}+{{c}^{2}}-{{a}^{2}}=8\)，则\(\triangle ABC\)的面积为________．