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笔者在研究2010年四川省理科数学第20题时有一发现,现撰文如下。题目1已知定点A(-1,0),F(2,0),定直线l:x=1/2,不在x轴上的动点P与点F的距离是它到直线l的距离的2倍。设点P的轨迹为曲线E,过点F的直线交曲线E于B、C两点,直线AB、AC分别交l于点M、N。(1)求曲线E的方程;(2)试判断以线段MN为直径的圆是否过点F,并说明理由。本题的第一问得出的结论是一条双曲线,而第二问的结论是肯定的。现将本题的条件一般化,可得出双曲线的一个结论。
When I was studying the 20th issue of Science Mathematics in Sichuan Province in 2010, I found that I now write as follows. Problem 1 Known fixed point A (-1,0), F (2,0), fixed straight line l: x = 1/2, not on the x-axis of the moving point P and its distance F is its distance from the straight line l 2 times. Set point P trajectory for the curve E, F over the point of the straight line to the curve E at B, C two points, AB, AC line, respectively, to pay l point M, N. (1) find the equation of curve E; (2) try to judge whether the circle with line segment MN as the diameter crosses F, and explain the reason. The first question from this question concludes with a hyperbola, while the second question concludes with a positive one. Now the general conditions of this question, can draw a conclusion hyperbola.