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一道解析几何问题,给笔者留下了深刻的印象,笔者借助几何画板,对其进行深入探究,得到圆锥曲线的一组有趣命题,以飨读者.题目如图,已知椭圆C∶x2/2+y2=1的左、右焦点分别为F1,F2,下顶点为A,点P是椭圆上任一点,圆M是以PF2为直径的圆.(1)当圆M的面积为π8,求PA所在的直线方程;(2)当圆M与直线AF1相切时,求圆M的方程;(3)求证:圆M总与某个定圆相切.
A parse the geometric problems left a deep impression on the author, the author with the help of geometric drawing board, in-depth exploration of them, get a series of interesting conical curve proposition, to readers. The subject is shown, the oval C: x2 / 2 The left and right focal points of + y2 = 1 are respectively F1 and F2, the lower vertex is A, point P is any point on the ellipse, and circle M is a circle of diameter PF2. (1) When the area of circle M is π8, find PA Where the line equation; (2) when the circle M tangent with the straight line AF1, find the equation of the circle M; (3) verify: the circle M with a fixed circle tangent.