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).當(dāng)時(shí)，求點(diǎn)的坐標(biāo)；

 

 (2).當(dāng)為何值時(shí)，四邊形的兩條對(duì)角線互相垂直；

(3).猜想線段與之間的數(shù)量關(guān)系，并證明你的結(jié)論．

 

Answer 1

【答案】

解：（1）點(diǎn)在拋物線上，且，，······························ 1分

    點(diǎn)與點(diǎn)關(guān)于軸對(duì)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(duì)角線互相垂直時(shí)，由對(duì)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(duì)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(duì)稱性得點(diǎn)，··················································· 9分

    ，

    ．                                                        10分

 

【解析】略