Question

正数x,y满足x2－y2＝2xy,求(x-y)/(X+Y）的值

Answer 1

正数x,y满足x²-y²＝2xy,求(x-y)/(X+Y）的值
解：将方程x²-y²=2xy的两边同除以xy得：(x/y)-(y/x)=2............(1)
令x/y=u，则y/x=1/u，代入(1) 式得u-(1/u)=2，即有u²-2u-1=0，故u=(2+√8)/2=1+√2
于是得(x-y)/(x+y)=[(x/y)-1]/[(x/y)+1]=(u-1)/(u+1)=(√2)/(2+√2)=(√2)(2-√2)/2=(2√2-2)/2=√2-1.

Answer 2

x2-2xy+Y2－2*y2=0
(X-Y)²-2Y²=0
X=(1-根号2）因为为正数，所以不合题意，舍去
X=(1+根号2）Y
(x-y)/(X+Y)=根号2-1

Answer 3

x^2 - y^2 = 2xy
x/y - y/x = 2
(x/y - y/x)^2 = 4
(x/y + y/x)^2 = 8
x/y+y/x = 2*2^(1/2)

(x-y)/(x+y) = (x-y)^2/(x^2-y^2) = (x^2 - 2xy + y^2)/(2xy)
= (1/2)(x/y +y/x) - 1
=2^(1/2) -1

Answer 4

根号2-1