已知曲线\(C\)的参数方程为\(\begin{cases} & x=2\cos \theta \\ & y=\sin \theta \end{cases}(\theta \)为参数\()\)，以坐标原点\(O\)为极点，\(x\)轴的正半轴为极轴建立极坐标系，直线\(l\)的极坐标方程为\(\rho \sin \left( \dfrac{\pi }{4}+\theta \right)=2\sqrt{2}\)．

\((1)\)将曲线\(C\)上各点的纵坐标伸长为原来的两倍，得到曲线\({{C}_{1}}\)，写出曲线\({{C}_{1}}\)的极坐标方程；

\((2)\)射线\(\theta =\dfrac{\pi }{6}\)与\({{C}_{1}},l\)的交点分别为\(A,B\)，射线\(\theta =-\dfrac{\pi }{6}\)与\({{C}_{1}},l\)的交点分别为\({{A}_{1}},{{B}_{1}}\)，求\(\Delta OA{{A}_{1}}\)与\(\Delta OB{{B}_{1}}\)的面积之比．