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在f(x,y)=0的条件下,求u=g(x,y)的最值,我们称这类问题为解析型最值问题,其中把f(x,y)=0视为定曲线,u=g(x,y)视为动曲线,在中学阶段解这类问题,往往都是借助于一些特殊的方法,学生不易掌握,本文给出一种极坐标解法,供读者参考。 例1 实数x、y满足4x~2-5xy+4y~2=5,又设S=x~2=y~2,则(1993年全国高中数学联赛试题) 解:定曲线可化为p~2=10/8-5sin2θ 当sin2θ=1时,p_(max)~2=10/3; 当sin2θ=-1时,p_(min)~2=10/13. 而动曲线S=x~2+y~2=p~2,
Under the condition of f(x,y)=0, find the maximum value of u=g(x,y). We call this kind of problem an analytical-valued problem, where f(x,y)=0 is treated as Curve, u = g (x, y) as a moving curve, in the middle school to solve these problems, often by means of some special methods, students are not easy to master, this paper gives a polar coordinate solution for readers to reference . Example 1 The real numbers x and y satisfy 4x~2-5xy+4y~2=5, and S=x~2=y~2, then (1993 National High School Mathematics League Questions) Solution: The curve can be changed to p~ 2=10/8-5sin2θ When sin2θ=1, p_(max)~2=10/3; When sin2θ=-1, p_(min)~2=10/13. And the dynamic curve S=x~2 +y~2=p~2,