当点A（1,2）B（3,1）C（2,3）到直线x-my=0的距离的平方和取最大值时,m等于

题目

当点A（1,2）B（3,1）C（2,3）到直线x-my=0的距离的平方和取最大值时,m等于

答案

d=(1-2m)²/(m²+1)+(3-m)²/(m²+1)+(2-3m)²/(m²+1)
=(4m²-4m+1+m²-6m+9+9m²-12m+4)/(m²+1)
=(14m²-22m+14)/(m²+1)
14m²-22m+14=dm²+d
(14-d)m²-22m+14-d=0
484-4(14-d)²>=0
(d-14)²<=121
-11<=d-14<=11
最大=25
(14m²-22m+14)/(m²+1)=25
11m²+22m+11=0
m=-1