椭圆伸缩变换的性质与两道高考题的简便解法

• 作 者: 齐昊鹏
• 机 构: 山东省寿光中学高二15班,
• 出 处: 数学通讯
• 发表日期: 2015年17期
• 关键词: 椭圆变换 高考题 中学数学 平面直角坐标系 平行性 离心率 段长度 公共点 椭圆问题 说明理由

下面先给出椭圆伸缩变换的有关结论:定理已知椭圆C:x2/a2+y2/b2=1(a>b>0),经过变换: x′=x/a ,y′=y/b,椭圆C变为圆C′:x′2+y′2=1,在这个变换下,变换前、后的图形具有如下性质:(1)点的位置不变,即椭圆C上的点经变换后都在圆C′上;(2)直线的平行性保持不变且相互平行(或在一条直线上)的线段长度的比也不变;

The following is the first conclusion about the elliptic expansion and contraction transformation: Theorem It is known that the ellipse C: x2 / a2 + y2 / b2 = 1 (a> b> 0) transformed by: x ’= x / a, y’ = y / b , The ellipse C becomes a circle C ’: x’2 + y’2 = 1. Under this transformation, the figures before and after transformation have the following properties: (1) the position of the point is constant, that is, the point on the ellipse C (2) the straight line parallelism remains unchanged and parallel to each other (or in a straight line) the ratio of the length of the line segment does not change;