Question

已知直線l₁:ax+2y+6=0和直線l₂:x+(a-1)y+a²-1=0,

（1）試判斷l(xiāng)₁與l₂是否平行；

（2）l₁⊥l₂時(shí)，求a的值.

Answer 1

（1）a=-1時(shí)，l₁∥l₂（2）a=

解析:

（1）方法一  當(dāng)a=1時(shí)，l₁:x+2y+6=0,

l₂:x=0,l₁不平行于l₂;

當(dāng)a=0時(shí)，l₁:y=-3,

l₂:x-y-1=0,l₁不平行于l₂;                                                                                                                         

當(dāng)a≠1且a≠0時(shí)，兩直線可化為

l₁:y=--3,l₂:y=-(a+1),

l₁∥l_2，解得a=-1,                                                                                                         

綜上可知，a=-1時(shí)，l₁∥l₂，否則l₁與l₂不平行.                                                                                              

方法二  由A₁B₂-A₂B₁=0，得a（a-1）-1×2=0，

由A₁C₂-A₂C₁≠0，得a(a²-1)-1×6≠0,                                                                                                           

∴l(xiāng)₁∥l₂                                                                                                                              

a=-1,                                                                                                                               

故當(dāng)a=-1時(shí)，l₁∥l₂，否則l₁與l₂不平行.                                                                                     

（2）方法一  當(dāng)a=1時(shí)，l₁:x+2y+6=0,l₂:x=0,

l₁與l₂不垂直，故a=1不成立.                                                                                                                               當(dāng)a≠1時(shí)，l₁:y=-x-3,

l₂:y=-(a+1),                                                                                                                                                由·=-1a=.                                                                                                                        

方法二  由A₁A₂+B₁B₂=0，得a+2(a-1)=0a=.