解:该直线方程是: y-1=k(x-2)
那么,该直线分别交x轴和y轴的交点是:
y=0,-1=kx-2k,x=(2k-1)/k 即A((2k-1)/k,0)
x=0,y-1=-2k,y=1-2k 即B(0,1-2k)
S△AOB=(1/2)(1-2k)(2k-1)/k
=(1/2)(2k-1-4k^2+2k)/k=(1/2)(4k-1-4k^2)/k
=2-[1/(2k)]-2k
S'=[1/(2k^2)]-2=0,k^2=4,k=±2(正值不合),所以k=-2
A(2.5,0) B(0,5)
Smin=(1/2)×2.5×5=6.25
直线L的方程是: y-1=-2(x-2)