题目

Naps—short sleeps during the day—may improve memory, experts said recently. New research conducted by brain researcher Avi Karni of the University of Haifa in Israel explores the possibility that naps help lock in sometimes long-term memories. “We still don't know exactly how memory system works during sleep, but the results of this research suggest it is possible to speed up memory consolidation(巩固),” Karni said. Long-term memory refers  to memories that stay with us for years, such as “what” memories -- a car accident that happened yesterday-- or “how to” memories, such 如图，已知直线，它与轴、轴的交点分别为A、B两点． （1）求点A、点B的坐标； （2）设F是轴上一动点，用尺规作图作出⊙P，使⊙P经过点B且与轴相切于点F（不写作法和证明，保留作图痕迹）； （3）设（2）中所作的⊙P的圆心坐标为P（），求与的函数关系式； （4）是否存在这样的⊙P，既与轴相切又与直线相切于点B，若存在，求出圆心P的坐标；若不存在，请说明理由．