X²-Y²=2XY,求(X-Y)÷(X+Y)
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X²-Y²=2XY,求(X-Y)÷(X+Y)
![X²-Y²=2XY,求(X-Y)÷(X+Y)](/uploads/image/z/19837461-21-1.jpg?t=X%26sup2%3B-Y%26sup2%3B%3D2XY%2C%E6%B1%82%28X-Y%29%C3%B7%28X%2BY%29)
∵ x²-y²=2xy,
∴(x-y)(x+y)=2xy,
∴(x-y)/(x+y)
=(x-y)²/(x²-y²)
=(x²-2xy+y²)/2xy
=[x²+y²-(x²-y²)]/2xy
=2y²/2xy
=y/x
又由x²-y²=2xy,
得x/y-y/x=2,
即(y/x)²+2(y/x)-1=0
∴y/x=-1+√2或y/x=-1-√2
(舍去,因为X,Y都是正数).
即(x-y)/(x+y)
=√2-1
∴(x-y)(x+y)=2xy,
∴(x-y)/(x+y)
=(x-y)²/(x²-y²)
=(x²-2xy+y²)/2xy
=[x²+y²-(x²-y²)]/2xy
=2y²/2xy
=y/x
又由x²-y²=2xy,
得x/y-y/x=2,
即(y/x)²+2(y/x)-1=0
∴y/x=-1+√2或y/x=-1-√2
(舍去,因为X,Y都是正数).
即(x-y)/(x+y)
=√2-1
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