已知X²+Y²+X²Y²-4XY+1=0 求:(X-Y)^2008-(XY)^2008
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![已知X²+Y²+X²Y²-4XY+1=0 求:(X-Y)^2008-(XY)^2008](/uploads/image/z/1644761-65-1.jpg?t=%E5%B7%B2%E7%9F%A5X%26%23178%3B%2BY%26%23178%3B%2BX%26%23178%3BY%26%23178%3B-4XY%2B1%3D0+%E6%B1%82%3A%EF%BC%88X-Y%29%5E2008-%28XY%29%5E2008)
已知X²+Y²+X²Y²-4XY+1=0 求:(X-Y)^2008-(XY)^2008
已知X²+Y²+X²Y²-4XY+1=0 求:(X-Y)^2008-(XY)^2008
已知X²+Y²+X²Y²-4XY+1=0 求:(X-Y)^2008-(XY)^2008
X²+Y²+X²Y²-4XY+1=0
(X-Y)^2+(XY-1)^2=0
所以X-Y=0 XY-1=0
(X-Y)^2008-(XY)^2008
=0^2008-1^2008
=0-1
=-1
上式可变换为
(x-y)^2+(xy-1)^2=0
所以
x-y=0
xy-1=0
所以X=Y=1
:(X-Y)^2008-(XY)^2008=-1
先将Y²+X²+X²Y²-4XY+1=0化简为(Y-X)²+(XY-1)²=0即可得Y-X=0 、XY-1=0可以算出X=Y=±1再将结果代入(X-Y)^2008-(XY)^2008=-1
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