Question

已知一次函数y=kx+b的图像与反比例函数y=-8/x的图像相交于A,B两点,其中A点的横坐标与B点的纵坐标均为2

求在坐标轴上是否存在一点P使△OAP为等腰三角形？（P点有8个，请写明步骤，谢谢）

Answer 1

A(2, -4)
OA² = (2 - 0)² + (-4 - 0)² = 20
(a) P在x轴上, P(p, 0)
(i) OA = AP
(2 - p)² + (-4 - 0)² = 20
p = 4 (舍去p = 0)
P(4, 0)
(ii) OA = OP
20 = p²
p = ±2√5
P(2√5, 0), P(-2√5, 0)
(iii) OP = AP
p² = (2 - p)² + (-4 - 0)²
p = 5
P(5, 0)

(b) P在y轴上, P(0, p)
(i) OA = AP
(2 - 0)² + (-4 - p)² = 20
p = -8 (舍去p = 0)
P(0, -8)
(ii) OA = OP
p² = 20
p = p = ±2√5
P(0, 2√5), P(0, -2√5)
(iii)OP = AP
p² = (2 - 0)² + (-4 - p)²
p = -5/2
P(0, -5/2)