1,已知1/x + 1/y = 1/(x+y)则 y/x + x/y = _________2,已知a + 1/a = 7则 a - 1/a = ______________
问题描述：
1,已知1/x + 1/y = 1/(x+y)
则 y/x + x/y = _________
2,已知a + 1/a = 7
则 a - 1/a = ______________
答
1题 把(x+y)乘到左边，化简可得y/x+x/y=1
2题 设a-1/a=n 与a+1/a=7解方程组
得a=(7+n)/2
1/a=(7-n)/2
二者互为倒数，即(7+n)/2=2/(7-n)
a-1/a=n=3√5 (√是根号)
答
1,
1/x + 1/y = 1/(x+y)
(x+y)/xy=1/(x+y)
(x+y)^/xy=1
(x^2+2xy+y^2)/xy=1
(x^2+y^2)/xy+2=1
x^2/xy+y^2/xy=-1
x/y+y/x=-1
2,
a - 1/a =√(a-1/a)^2=√(a^2+1/a^2-2)
=√(a^2+2+1/a^2-4)
=√((a+1/a)^2-4)
=√(7^2-4)
=√(49-4)
=√45
=3√5
答
(2表示平方)
y/x+x/y=(x2+y2)/xy=(x2+2乘以xy+y2)/xy-(2乘以xy)/xy=((x+y)2)/xy-2
因为1/y+1/x=1/(x+y)化简成(x+y)2=xy
带入,答案为-1
（a+1）/a=1+(1/a)=7推出(1/a)=6
(a-1)/a=1-(1/a)=-5