x+y=20(x-10)²+(y-10)²=8 求x-y

x+y=20(x-10)²+(y-10)²=8 求x-y
x+y=20
(x-10)²+(y-10)²=8 求x-y

x+y=20(x-10)²+(y-10)²=8 求x-y
设 a=x-10,b=y-10
a+b = x+y - 20 = 0
(a+b)² = a²+b²+2ab=0
a²+b² = 8
∴2ab = - 8
(a-b)² = a²+b² - 2ab = 16
a-b = ±4
x-y = a-b = ±4

x+y=20,则x=20-y,代入下式
(x-10)²+(y-10)²=8
(-y+10)²+(y-10)²=8
2(y-10)²=8
(y-10)²=4
y=12或8
x=8或12
x-y=8-12=-4或x-y=12-8=4

(x-10)²+(y-10)²=x^2-20x+100+y^2-20y+100=x^2+y^2-20(x+y)+200=8
因为x+y=20

代入得x^2+y^2-20*20+200=8
所以x^2+y^2=208
由x+y=20两边平方得x^2+y^2+2xy=400
所以2xy=192
所以(x-y)^2=x^2+y^2-2xy=208-192=16
所以x-y=4

+4或-4