Question

如圖，點(diǎn)在拋物線上，過點(diǎn)作與軸平行的直線交拋物線于點(diǎn)，延長分別與拋物線相交于點(diǎn)，連接，設(shè)點(diǎn)的橫坐標(biāo)為，且．

(1). （4分） 當(dāng)時(shí)，求點(diǎn)的坐標(biāo)；
(2). （2分）當(dāng)為何值時(shí)，四邊形的兩條對角線互相垂直；
(3). （4分） 猜想線段與之間的數(shù)量關(guān)系，并證明你的結(jié)論．

Answer 1

（1）解：（1）點(diǎn)在拋物線上，且，，······························ 1分
點(diǎn)與點(diǎn)關(guān)于軸對稱，．························································ 2分
設(shè)直線的解析式為，
．······················································································· 3分
解方程組，得．································································· 4分
（2）當(dāng)四邊形的兩對角線互相垂直時(shí)，由對稱性得直線與軸的夾角等于所以點(diǎn)的橫、縱坐標(biāo)相等，      5分
這時(shí)，設(shè)，代入，得，．
即當(dāng)時(shí)，四邊形的兩條對角線互相垂直．········································· 6分
（3）線段．········································································································ 7分
點(diǎn)在拋物線，且，
得直線的解析式為，
解方程組，得點(diǎn)······················································· 8分
由對稱性得點(diǎn)，··················································· 9分
，
．                                                      10分

解析