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，求點(diǎn)的坐標(biāo)；

 

 (2).當(dāng)為何值時，四邊形的兩條對角線互相垂直；

(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

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，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，設(shè)，代入，得，．

   即當(dāng)時，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:略