双曲线y=k x经过直角梯形ABEO顶点E,BA⊥x轴于点A，BE//OA，双曲线与AB相交于点D

Question

，与OB相交于点P，若OP=2BP,且四边形DBEO的面积为20，则K=

Answer 1

设A(a, 0), B(a, b), 则D(a, k/a), E(k/b, b)
AB的方程: y = bx/a; 与y = k/x联立, x = √(ka/b) (舍去负值)
P(√(ka/b), √(kb/a)

四边形DBEO的面积 = 直角梯形ABEO的面积 - ∆OAD的面积
= (1/2)(EB + OA)*AB - (1/2)*OA*AD
= (1/2)(a - k/b + a)b - (1/2)*a*k/a
= (1/2)(2ab - k -k)
= ab - k = 20
ab = 20 + k (i)
OP=2BP
OP = (2/3)OB
P的横坐标 = (2/3)*B的横坐标
√(ka/b) = 2a/3
ab = 9k/4 (ii)
20 + k = 9k/4
k = 16