若﹛（x,y)|ax+y-b=0﹜∩﹛(x,y)|x+ay+1=0﹜=空集,则求a,b应满足什么条件.a=1且b≠-1或a=-1且b≠1求详细过程谢谢!

Question

若﹛（x,y)|ax+y-b=0﹜∩﹛(x,y)|x+ay+1=0﹜=空集,则求a,b应满足什么条件.
a=1且b≠-1或a=-1且b≠1求详细过程谢谢!

Answer 1

若{(x,y)|ax+y-b=0} ∩{(x,y)|x+ay+1=0}=空集,则:
ax+y-b=0------①
x+ay+1=0------②
①∩②=空集说明方程①②没有公共根
①-②得:
(a-1)x+(1-a)y-(b+1)=0
要使得该方程无解,显然,当a=1且b≠-1时满足条件(a=-1时,代入方程(a+1)x+(1+a)y-(1+b)=0得:1+b=0,要使得方程无解,此时b≠-1时满足条件)
①+②得：
(a+1)x+(1+a)y+(1-b)=0
要使得该方程无解,显然,a=-1时,代入得1-b=0,即b≠1时,该方程无解
∴a,b满足的条件是a=1且b≠-1或a=-1且b≠1