Exploration of fixed length problems and reconstructed question types in conic sections: A teaching case of question 18 in the 2025 new college entrance examination volume I
圆锥曲线中求定长问题及重构题型探究——以2025年新高考Ⅰ卷18题教学为例

摘要

圆锥曲线定长问题是高考解析几何压轴题的核心考查方向，其难点在于如何在复杂几何条件与繁复代数运算之间建立高效转换通道。本文以2025年新高考Ⅰ卷18题为研究载体，首先展示原题的完整代数翻译过程，继而通过将定长条件一般化为参数t、将斜率倍数条件一般化为参数q，重构出更具普遍性的题型框架。在引申问题（i）中，本文分别运用坐标代数法、向量共线法与相似三角形几何法三种路径求解动点R坐标，对比发现向量法在运算简洁性与逻辑清晰度上最优。在引申问题（ii）中，推导得出点P的轨迹为以Q(0, -(2+qt-t)/2)为圆心的圆，进而将|PM|最大值问题转化为椭圆上点到圆心Q的最大距离与圆半径之和，最终给出含参表达式。研究表明，斜率倍数q通过约束P点位置间接影响|PM|的最值，二者呈正相关趋势。本文提炼的“条件翻译分层法”和“动点来源追溯法”可为高三解析几何复习提供可操作的教学策略。

Abstract

Fixed length problems of conic sections constitute a core examination focus of the final challenge analytic geometry questions in the college entrance examination. Their difficulty lies in building an efficient conversion channel between complex geometric conditions and cumbersome algebraic operations. Taking Question 18 of the 2025 New College Entrance Examination Volume I as the research carrier, this paper first demonstrates the complete algebraic translation process of the original problem. Then, by generalizing the fixed length condition to parameter t and the slope multiple condition to parameter q, a more universal question type framework is reconstructed. In extended problem (i), three approaches, namely the coordinate algebra method, the vector collinearity method and the similar triangle geometric method, are adopted to solve the coordinates of moving point R. Comparison reveals that the vector method achieves optimal performance in computational simplicity and logical clarity. In extended problem (ii), it is deduced that the trajectory of point P is a circle with the center Q(0,−（2+qt−t）/2）. Furthermore, the maximum value problem of|PM|is transformed into the sum of the maximum distance from a point on the ellipse to center Q and the radius of the circle, and the final parametric expression is obtained. The study shows that the slope multiple q indirectly affects the maximum and minimum values of |PM|by restricting the position of point P, and there exists a positive correlation between them.The “hierarchical condition translation method” and “moving point source tracing method” summarized in this paper provide practical teaching strategies for senior three review of analytic geometry.