已知X.Y是实数且满足X^2+XY+Y^2-2=0,设M=X^2-XY+Y^2,则M的取值范围是
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已知X.Y是实数且满足X^2+XY+Y^2-2=0,设M=X^2-XY+Y^2,则M的取值范围是
X^2+Y^2=2-XY
X^2+Y^2>=2XY
所以2-XY>=2XY
XY=-2/3
M=X^2+Y^2-XY=X^2+XY+Y^2-2XY=2-2XY>=2+2*(-2/3)=2/3
即M>=2/3
又
(X+Y)^2>=0
X^2+Y^2>=-2XY
2-XY>=-2XY
XY>=-2
-2XY
X^2 + XY + Y^2 - 2 = 0
则有:(X - Y)^2 + 3XY - 2 = 0
3XY = 2 - (X - Y)^2 <= 2
XY <= 2/3
-2XY >= -4/3
M=X^2 - XY + Y^2 = X^2 + XY + Y^2 - 2XY = 2 - 2XY >= 2 - 4/3 = 2/3
所以,M >= 2/3...
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X^2 + XY + Y^2 - 2 = 0
则有:(X - Y)^2 + 3XY - 2 = 0
3XY = 2 - (X - Y)^2 <= 2
XY <= 2/3
-2XY >= -4/3
M=X^2 - XY + Y^2 = X^2 + XY + Y^2 - 2XY = 2 - 2XY >= 2 - 4/3 = 2/3
所以,M >= 2/3
又有:(X + Y)^2 - XY - 2 = 0
XY = (X + Y)^2 - 2 >= -2
-2XY <= 4
所以 M= 2 - 2XY <= 2 + 4 = 6
则 M < = 6
综上,M的取值范围为: 2/3<= M <= 6
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