(1) x = p時(shí),y = (p - 2)(p - m) - (p - 2)(p - m) = 0,所以(p,0)在拋物線(xiàn)上.
(2) 設(shè)A(a,0),C(c,0)
x = a時(shí),y = (a - 2)(a - m) - (p - 2)(p - m) = 0 (1)
x = c時(shí),y = (c - 2)(c - m) - (p - 2)(p - m) = 0 (2)
(1)-(2):(a - 2)(a - m) - (c - 2)(c - m) = 0
a² - (m+2)a + 2m = c² - (m+2)c + 2m
a² - c² = (m+2)a - (m+2)c
(a + c)(a - c) = (m + 2)(a - c)
根據(jù)題意,a,c不相等,a + c = m + 2
因?yàn)閽佄锞€(xiàn)與x軸只能有兩個(gè)交點(diǎn),(p,0)必為A或C.
(a) (p,0)為A
a = p
c = m + 2 - a = m + 2 - p
根據(jù)題意,a > c,m + 2 ≥ 2p
a -c = p - (m + 2 - p) = 2p - (m + 2) < 0
與a > c矛盾,舍去
(b) (p,0)為C
c = p
a = m + 2 - c = m + 2 - p
根據(jù)題意,a > c,m + 2 ≥ 2p
a - c = (m + 2 - p) - p = (m + 2) - 2p > 0
符合題意
|OC| = c = p
|OA| = a = m + 2 - p
(3)|OB| = |OC| = p
△AOB的面積S = (1/2)|OB|*|OA| = p(m+2-p)/2
= -[p - (m+2)/2]² + (m+2)²/8
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