已知实数x,y满足x2+y2=2x,求x2y2的范围

已知实数x,y满足x2+y2=2x,求x2y2的范围
已知实数x,y满足x2+y2=2x,求x2y2的范围

已知实数x,y满足x2+y2=2x,求x2y2的范围
∵x^2+y^2=2x ∴（x-1)^2+y^2=1
设x=1+cosa,y=sina 则
x^2y^2=(1+sina)^2(cosa)^2=(1+sina)^2[(1-(sina)^2]
=(1+sina)^3(1-sina)==(1+sina)(1+sina)(1+sina)(3-3sina)/3
≤[(1+sina+1+sina+1+sina+3-3sina)/4]^4/3
=(3/2)^4/3=27/16