Discussion on the application of parametric equations of lines in conic section problems of college entrance examinations
例谈直线的参数方程在高考圆锥曲线大题中的应用

摘要

本文旨在探讨直线的参数方程在解决高考圆锥曲线线段长度问题中的优势应用。文章首先从几何视角阐释了参数t的几何意义，奠定了方法基础。进而，通过精析2021年全国I卷、2019年全国I卷及2021年新高考II卷三道典型真题，具体演示了如何利用参数方程设线、联立，并借助参数t与韦达定理直接处理弦长及线段比例关系，从而规避复杂的坐标运算。案例分析表明，该方法在处理以定点为线段端点的长度问题时尤为高效，能显著简化计算流程。本文为高中师生攻克此类解析几何难题提供了一种清晰的解题视角和可操作的技术路径。

Abstract

This paper aims to explore the advantageous applications of the parametric equations of straight lines in solving problems related to the length of line segments in conic sections in the College Entrance Examination. The article begins by explaining the geometric significance of the parameter t from a geometric perspective, establishing the methodological foundation. Subsequently, by analyzing three typical exam questions—the 2021 National I Volume, the 2019 National I Volume, and the 2021 New College Entrance Examination II Volume—the paper demonstrates how to use parametric equations to establish lines and set up equations, and directly handle chord lengths and segment proportional relationships with the parameter t and Vieta's theorem, thereby avoiding complex coordinate calculations. The case analysis shows that this method is particularly efficient in dealing with length problems where fixed points serve as the endpoints of line segments, significantly simplifying the calculation process. This paper provides high school teachers and students with a clear problem-solving perspective and an operable technical approach to tackle such analytic geometry challenges.