我校高三2月入学试题(理科)16题

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已知过椭圆\(C\)：\(\dfrac{\textit{x}^2}{\textit{a}^2}+\dfrac{\textit{y}^2}{\textit{b}^2}=1(\textit{a}>\textit{b}>0)\)的焦点\(F\) 且倾斜角为\(45^\circ\)的直线交椭圆\(C\)于\(A\)、\(B\) 两点, \(\dfrac{1}{|AF|}+\dfrac{1}{|BF|}=2\textit{a}\), 若存在以\((0\),\(\textit{b})\)为圆心的圆与椭圆\(C\)有三个不同的公共点, 则\(\textit{ab}\)的取值范围为\(\underline{\qquad \blacktriangle\qquad }.\)