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我们知道圆O:x~2+y~2=r~2(r>0),它的外切矩形是正方形,由对称性可知:(1)正方形顶点的轨迹在以圆的圆心为圆心,2~(1/2)r为半径的圆周上;(2)S_(ABCD)=4r~2为定值。那么我们高中所学习过的三种圆锥曲线,它们的“外切”矩形的顶点和面积又会有怎样的性质呢?本文运用了数学中的类比推理方法来研究解析几何的有关问题,从熟悉的圆→椭圆→双曲线→抛物线,
We know that the circle O:x~2+y~2=r~2(r>0). Its circumscribed rectangle is a square. It can be known from the symmetry that: (1) The trajectory of a square vertex is centered on the center of the circle. 2~(1/2)r is a circle on the radius; (2) S_(ABCD)=4r~2 is a fixed value. So what kind of properties of the three types of conical curves that we have studied in high school and their vertices and areas of the “outer cut” rectangles? This paper uses analogy inference methods in mathematics to study the problems of analytical geometry. From the familiar circle → ellipse → hyperbola → parabola,