已知椭圆 C: x 2 a 2 + y 2 b 2 =1(a>b>0) 的离心率为 2 2 ,其左、右焦点分别为F 1 、F 2 ,点P是坐标平面内一点,且 |OP|= 7 2 , P F 1 ? P F 2 = 3 4 (O为坐标原点). (1)求椭圆C的方程; (2)过点 S(0,- 1 3 ) 且斜率为k的动直线l交椭圆于A、B两点,在y轴上是否存在定点M,使以AB为直径的圆恒过这个点?若存在,求出M的坐标和△MAB面积的最大值;若不存在,说明理由.
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