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一、提出问题＂数学教学应当从问题开始＂。教师首先提出问题:在椭圆x2/a2+y2/b2=1(a>b>0)上,哪个点到顶点B(0,b)的距离最远? 如图1所示＂度量＂(几何画板的菜单功能)BP的长,如|BP|=1.8cm。拖动点P,观察BP长的变化,很快发现,点P在哪里|BP|可以达到最大值。 有些同学发现:无论怎么拖动点P,|BP|最大时,点P的位置都不会位于x轴的上方!拖动点A,使椭圆很＂扁＂,这一结论仍然成立。 还有一部分同学以B为圆心,经过点P画圆,认为当这个圆把椭圆＂包＂起来时,或者说当这个圆与椭圆相切时,其半径长就是|BP|的最大值,如图2所示。
I. Asking questions “Mathematics teaching should begin with questions.” The teacher first asks the question: on the ellipse x2/a2+y2/b2=1 (a>b>0), which point is the farthest away from the vertex B(0,b)? Figure 1 shows the “measure” (geometry Sketchpad menu features) BP’s length, such as |BP|=1.8cm. Drag the point P, observe the change in BP length, and quickly discover where the point P is |BP| Some students found that: no matter how you drag the point P, when |BP| is maximum, the position of point P will not be located above the x axis! Dragging point A makes the ellipse “flat”, and this conclusion is still true. Some classmates use B as the center of the circle and draw a circle through P. They think that when the circle “wraps” the ellipse, or when the circle is tangent to the ellipse, the radius is the maximum of |BP|, such as Figure 2 shows.