Question

已知正数x，y满足x^2-y^2=2xy,求(x-y)\(x+y)的值？过程！！！

Answer 1

x-y)/(x+y)
（x-y)/(x+y)=（x-y)(x+y)/(x+y)^2=(x^2-y^2)/(x^2+2xy+y^2)
=2xy/(x^2+x^2-y^2+y^2)=y/x
x^2-y^2=2xy
=>
x/y-y/x=2
设x/y为a
所以a-1/a=2
=>
a^2-1=2a
=>a^2-2a-1=0
a=1+√2或a=√2-1
所以原式=√2-1
（另一个舍去，因为X，Y都是正数）

Answer 2

x^2-y^2=(x+y)(x-y)=2xy

求的是（x-y)/(x+y)还是x+y-y/x
是（x-y)/(x+y)的话：
（x-y)/(x+y)=（x-y)(x+y)/(x+y)^2=(x^2-y^2)/(x^2+2xy+y^2)

=2xy/(x^2+x^2-y^2+y^2)=y/x
x^2-y^2=2xy
=>
x/y-y/x=2
设x/y为a
所以a-1/a=2
=>
a^2-1=2a
=>a^2-2a-1=0
a=1+√2或a=1-√2（注：√2是根号2）
所以：y/x=1+√2或1-√2

Answer 3

x^2-2xy+y^2=2y^2,
(x-y)^2=2y^2,

x>0,y>0,
2xy>0,
x^2-y^2>0,
x>y,
x-y>0

x-y=√2y,

x+y=√2y+2y=(2+√2)y
(x-y)/(x+y)=√2y/[(2+√2)y]=√2/(2+√2)=√2(2-√2)/2=√2-1