設(shè)L的方程為x/a+y/b=1
∵p(1,4)在L上
∴1/a+4/b=1
∴b=4a/(a-1)
∴a+b=a+4a/(a-1)=(a²+3a)/(a-1)
(a-1)²+5(a-1)+4
=-------------------=(a-1)+4/(a-1)+5
a-1
≥2√(a-1)·√4/(a-1)
=2×2+5=9
此時(shí)a-1=4/(a-1),即a=3,b=6
∴L的方程是x/3+y/6=1即y=-2x+6.
(2)S⊿=1/2·ab=a/2·4a/(a-1)=2a²/(a-1)
2(a-1)²+4(a-1)+2
=-------------------
a-1
=2(a-1)+2/(a-1)+4
≥2×2√(a-1)×1/(a-1)+4
=8
此時(shí)a-1=1/(a-1),即a=2,b=8
L的方程是x/2+y/8=1即y=-4x+8