Locus and Optimization Problems in Lower and Higher Dimensions

In this paper we consider a locus problem that originated in a practice examination for admission to Chinese universities. We have made the problem more interesting and challenging by adding an optimization component. We also extend the locus and optimization problems to sphere and hyperspheres. The original problem looks simple and yet can prove challenging to many students. The solutions for locus in 2D are accessible to high school students. The solutions for maximizing the area or rectangular box using Lagrange multipliers is accessible to university students who have learned multi-variable calculus. Finally, optimization using Grobner bases can be understood by those graduate students who have grasped the concept.