(1)设A(0, a), B(b, a), C(b, 0)
反比例函数y=k/x与AB(y = a)的交点为E(k/a, a)
反比例函数y=k/x与BC(x = b)的交点为F(b, k/b)
AE = k/a, CF = k/b
S1 S2 = (1/2)*OA*AE (1/2)OC*CF
= (1/2)a*k/a (1/2)b*k/b)
= k/2 k/2
= k = 2
k = 2
(2)设A(0, 2), B(4, 2), C(4, 0)
E(k/2, 2), F(4, k/4)
AE = k/2, EB = 4 - k/2
CF = k/4, FB = 2 - k/4
四边形OAEF的面积=
矩形OABC的面积 - 三角形OCF的面积 - 三角形BEF的面积
= 4*2 - (1/2)OC*CF - (1/2)EB*FB
= 8 - (1/2)*4*k/4 - (1/2)(4 - k/2)(2 - k/4)
= 8 - k/2 -(1/2)(8 - k - k k²/8)
= 4 k/2 - k²/16
= 5 - (k - 4)²/16
k = 4时,四边形OAEF的面积最大,为5.
此时E(2, 2)