Question

如圖，點在拋物線上，過點作與軸平行的直線交拋物線于點，延長分別與拋物線相交于點，連接，設點的橫坐標為，且．

(1). （4分） 當時，求點的坐標；
(2). （2分）當為何值時，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量關系，并證明你的結論．

Answer 1

（1）解：（1）點在拋物線上，且，，······························ 1分
點與點關于軸對稱，．························································ 2分
設直線的解析式為，
．······················································································· 3分
解方程組，得．································································· 4分
（2）當四邊形的兩對角線互相垂直時，由對稱性得直線與軸的夾角等于所以點的橫、縱坐標相等，      5分
這時，設，代入，得，．
即當時，四邊形的兩條對角線互相垂直．········································· 6分
（3）線段．········································································································ 7分
點在拋物線，且，
得直線的解析式為，
解方程組，得點······················································· 8分
由對稱性得點，··················································· 9分
，
．                                                      10分

解析