若实数x,y满足x2+y2=4则2xy/x+y-2的取值范围为____

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/05/26 11:03:52

2xy/(x+y-2)
=(x²+y²+2xy-4)/(x+y-2)
=[(x+y)²-4]/(x+y-2)
=(x+y+2)(x+y-2)/(x+y-2)
=x+y+2
x²+y²=4是圆,以原点为圆心,半径为2
设x+y+2=a,为直线.只要求出a值的范围就行.
即截距2-a最小,
直线与圆相切的时候分别是最大最小值(A,B为交点)
A（-√2,-√2） B（√2,√2)
分别代入式子a最大=2+2√2,a最小=2-2√2

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