已知两直线l1:ax-by+4=0,l2:(a-1)x+y+b=0.求分别满足下列条件的a,b的值. (1)直线l1过点(-3,-1),并且直线l1与l2垂直; (2)直线l1与直线l2平行,并且坐标原点到l1,l2的距离相等.

问题描述:

已知两直线l1:ax-by+4=0,l2:(a-1)x+y+b=0.求分别满足下列条件的a,b的值.
(1)直线l1过点(-3,-1),并且直线l1与l2垂直;
(2)直线l1与直线l2平行,并且坐标原点到l1,l2的距离相等.

(1)∵l1⊥l2
∴a(a-1)+(-b)•1=0,即a2-a-b=0①
又点(-3,-1)在l1上,
∴-3a+b+4=0②
由①②得a=2,b=2.
(2)∵l1∥l2,∴

a
b
=1-a,∴b=
a
1−a

故l1和l2的方程可分别表示为:
(a-1)x+y+
4(a−1)
a
=0,(a-1)x+y+
a
1−a
=0,
又原点到l1与l2的距离相等.
∴4|
a−1
a
|=|
a
1−a
|,∴a=2或a=
2
3

∴a=2,b=-2或a=
2
3
,b=2.