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化简圆锥曲线的方程用坐标系的平移和旋转是众所周知的,然而由于圆锥曲线具有对称的特性,可以设想用轴对称变换的方法来达到此目的,它的基本思路是保持坐标系不动,选择适当的直线作对称轴,把曲线变成关于此直线对称的曲线,从而达到化简方程之目的。这是一种新的思路,所需要的基础知识反而比传统的方法更单纯,为达此目的,我们有以下三个定理,它们的证明并不困难,我们仅叙述之。定理1 设点P(x,y)和点P’(x’,y’)
Simplification of the equation of the conic section is well known by the translation and rotation of the coordinate system. However, because the conic curve has symmetry characteristics, it can be conceived to use axisymmetric transformation to achieve this purpose. Its basic idea is to keep the coordinate system immobile. Select the appropriate straight line as the axis of symmetry, and turn the curve into a symmetric curve about this line, so as to simplify the equation. This is a new way of thinking. The basic knowledge required is more simple than the traditional method. To achieve this goal, we have the following three theorems. Their proof is not difficult. We only describe it. Theorem 1 Set point P(x,y) and point P’(x’,y’)