直线L经过两条直线2x+y-8=0和x-2y+1=0的交点P,且与X轴的正半轴,Y轴的正半轴分别交于A,B两点

1. 二直线联立,可得P(3, 2)
设L的斜率为k, 方程y - 2 = k(x - 3), k < 0
分别取y = 0, x = 0, 可得A(3 - 2/k, 0), B(0, 2 - 3k)
S = (1/2)OA*OB = (1/2)(3 - 2/k)(2 - 3k) = 16
9k² + 20k + 4 = (9k + 2)(k + 2) = 0
k = -2/9或k = -2 (此为x - 2y + 1 = 0,舍去)
y = -2x/9 + 8/3
2.
k < 0
S = (1/2)OA*OB = (1/2)(3 - 2/k)(2 - 3k)
= (12 - 9k - 4/k)/2
≥ 6 + √[(-9k)(-4/k)]
= 6 + 6 = 12
此时-9k = -4/k
k² = 4/9
k = -2/3 (舍去k = 2/3 > 0)
y = -2x/3 + 4
(3)
PA² = (3 - 2/k - 3)² + (0 - 2)² = 4 + 4/k²
PB² = (3 - 0)² + (2 - 2 + 3k)² = 9 + 9k²
PA²*PB² = 36(1 + 1/k²)(1 + k²)
= 36(2 + k² + 1/k²)
≥ 36[2 + 2√(k² *1/k²)]
= 144
k² = 1/k²时, PA*PB最小, 此时k² = 1/k², k² = 1, k = -1 (舍去k = 1 > 0)
y = -x + 5