已知X,Y为正数,且(X²-Y²)/XY=2,求(X-Y)/(X+Y)的值
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已知X,Y为正数,且(X²-Y²)/XY=2,求(X-Y)/(X+Y)的值
已知X,Y为正数,且(X²-Y²)/XY=2,求(X-Y)/(X+Y)的值
已知X,Y为正数,且(X²-Y²)/XY=2,求(X-Y)/(X+Y)的值
(X²-Y²)/XY=2
则X²-Y²=2XY
则X²-Y²-2XY=0
X²-2XY+Y²-2Y²=0
(X-Y)²-2Y²=0
(X-Y+√2*y)(X-Y-√2*Y)=0
X=Y-√2*y 或 X=Y+√2*y
因为X,Y为正数
所以X=Y+√2*y
(X-Y)/(X+Y)=√2-1
(X²-Y²)/XY=2
即 x^2-2xy-y^2=0
x=(1+√2)y
代人得(X-Y)/(X+Y)=(1+√2-1)/(1+√2+1)=√2-1
原式=x/y-y/x=2
把x/y看做a,那么原式变为a-1/a=2
同时乘以a(x,y均大于0),解方程,得到两根,负根舍去
x/y=1+根2,x=1+根2y,代入
(X-Y)/(X+Y)=根2-1