Multidimensional Exploration of the Maximum and Minimum Problems Between Lines and Circles -- Taking the 12th question of the Science Mathematics section in the National Volume A of the 2024 College Entrance Examination as an example

Abstract

The maximum and minimum values of lines and circles are high-frequency examination points in the analytic geometry section of
the college entrance examination, often combined with cross-module comprehensive questions involving sequences. In this paper, the research
object is the 12th question of Science in the national first volume in 2024, which analyzes the proposition background, literacy orientation
and examination characteristics of the questions; Combining with the source of the textbook and the comparison of the real problems over the
years, this paper combs the inheritance and evolution trend of the proposition, and then constructs six solutions from the perspectives of geometric intuition, algebraic operation, parameter equation, etc., showing the diversity of problem-solving ideas. At the same time, the author
carried out a variety of inquiry to explore the nature of the problem, and combined with the teaching practice, gave teaching suggestions from
three aspects: cross module integration, multiple solutions to a problem, and the combination of number and shape, providing reference for the
preparation for the college entrance examination and the cultivation of students' mathematical thinking.