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研究圆锥曲线性质是中学解析几何的一个重要任务,传统方法是对每种曲线作孤立地研究的,但如应用对偶法则,可给出一种统一的研究方法. 设椭圆b2x2+a2y2=a2b2,取对偶法则f:a←→a,b←→bi,便可得双曲线b2x2-a2y2=a2b2,不难验证,运用对偶法则作类比推理,两种曲线的许多基本性质有一种对应关系,这启示我们,不妨设有心曲线方程为Ax2+By2=1,可先作统一研究,然后再分类讨论之.
The study of the properties of conic curves is an important task for analytical geometry in middle schools. The traditional method is to study each curve in isolation. However, if the dual law is applied, a unified research method can be given. Set the ellipse b2x2+a2y2=a2b2, Take the duality law f:a←→a,b←→bi to obtain the hyperbola b2x2-a2y2=a2b2. It is not difficult to verify that the dual rule is used for analogy reasoning. Many basic properties of the two curves have a corresponding relationship. Revelation, we may wish to set the curve of the heart as Ax2+By2=1, we can do a unified study first, and then discuss the classification.