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《数学通报》1981年第1期,刊登了李希颜同志的《二次曲线离心率e与曲线形状的关系》,该文以抛物线为例证明了“离心率e相等的二次曲线一定相似”。那么相似的二次曲线离心率是否相等呢?本文将用一个既统一又简单的方法证明下面的定理。为了节省篇幅,有关图形相似的概念,就不再重述了。定理:两二次曲线相似的充要条件是这两条曲线的离心率相等。设两二次曲线c_1、c_2的离心率分别为e_1、e_2,即要证明c_1∽c_2(?)e_1=e_2。证明:不失一般性,将c_1、c_2置于同一极座标系中(如图),任作一射线OA,分别与c_1、c_2相交于
“Mathematical Bulletin,” No. 1, 1981, published Comrade Li Xiyan’s “Eccentricity of the second-order curve e and the relationship between the shape of the curve,” which uses a parabola as an example to prove that “the eccentricity of the eccentricity e equal to the quadratic curve must be similar.” So whether the eccentricity of similar quadratic curves are equal? In this paper, the following theorem will be proved by a unified and simple method. In order to save space, the concept of similar graphics will not be repeated. Theorem: The necessary and sufficient condition for the similarity of the two quadratic curves is that the eccentricity of the two curves is equal. Let the eccentricity of the two second-order curves c_1 and c_2 be e_1 and e_2 respectively, that is, c_1∽c_2(?)e_1=e_2. Proof: Without loss of generality, c_1 and c_2 are placed in the same polar coordinate system (pictured), and they act as a ray OA, intersecting with c_1 and c_2 respectively.