一道解析几何题的十五种解法赏析

Appreciation of Fifteen Solutions to an Analytic Geometry Problem

摘要: 解析几何的难点就是“算”，有效运算、简便运算求解析几何问题必须环环重视。考虑要如何设元，如何设方程，如何整体代换，如何进行转化化简。解析几何需要灵活运用条件转化多视角思考，本文对2020年全国卷1解析几何解法进行15种路径探析，主要围绕两条主线来展开，第一、设线法，针对多条动直线，寻找主动直线，设直线方程处理问题。第二、设点法，找消元等式，针对动点问题、寻找主要动点，用参数表示其他点。此问题入口宽、方法灵活多样，本文对第二问定点问题探索出十五种解法。不同的路径、运算量是不一样的，本题还呈现出不对称的韦达特征，本文实现了点代入曲线方程将不对称转化为对称问题；用根的和积关系转化消元；利用第三定义实现对称转化；出现斜率的积和关系用齐次化、点乘双根法。以及极点极线、二次对合、蝴蝶定理、参数方程、曲线系等解决问题。在圆锥曲线问题中要启发学生多角度，多层次思考问题，审时度势化简问题，举一反三提升学生数学核心素养。解析几何有效检测学生的直观想象、数学运算、逻辑推理等数学核心素养，基于“一核四层四翼”要注意思想方法的渗透，落实学科的立德树人。

Abstract: The difficulty of analytic geometry is “calculation”. Effective calculations and simple calculations must be taken seriously to solve analytic geometry problems. Consider how to set up elements, how to set up equations, how to substitute as a whole, and how to carry out transformation and simplification. Analytical geometry requires flexible use of conditions and multi-perspective thinking. This article analyzes 15 paths of analytic geometry solutions for National Volume 1 in 2020, mainly focusing on two main lines. First, the line setting method, looking for active lines for multiple moving straight lines. Straight line set the equation of straight line to solve the problem. Second, set point method, find elimination equations, find the main moving points for moving point problems, and use parameters to represent other points. This problem has a wide entrance and flexible and diverse methods. This article explores fifteen solutions to the second fixed-point problem. Different paths and calculation quantities are different. This problem also shows asym-metric Vedic characteristics. This article realizes the point substitution into the curve equation to transform the asymmetry into a symmetric problem; uses the sum-product relationship of the roots to transform the elimination; uses the third the definition implements symmetric transformation; the product-sum relationship where the slope appears uses the homogenization and dot product double root method. As well as solving problems such as poles and lines, quadratic involution, butterfly theorem, parametric equations, and curve systems. In the conic section problem, students should be inspired to think about the problem from multiple angles and levels, assess the situation and simplify the problem, and draw inferences from one example to improve students’ core mathematical literacy. Analytical geometry effectively tests students’ core mathematical qualities such as intuitive imagination, mathematical operations, and logical reasoning. Based on the “one core, four layers, and four wings”, we must pay attention to the penetration of ideological methods and implement the discipline’s moral education.