Question

如图，点在抛物线上，过点作与轴平行的直线交抛物线于点，延长分别与抛物线相交于点，连接，设点的横坐标为，且．

(1). （4分） 当时，求点的坐标；
(2). （2分）当为何值时，四边形的两条对角线互相垂直；
(3). （4分） 猜想线段与之间的数量关系，并证明你的结论．

Answer 1

（1）解：（1）点在抛物线上，且，，······························ 1分
点与点关于轴对称，．························································ 2分
设直线的解析式为，
．······················································································· 3分
解方程组，得．································································· 4分
（2）当四边形的两对角线互相垂直时，由对称性得直线与轴的夹角等于所以点的横、纵坐标相等，      5分
这时，设，代入，得，．
即当时，四边形的两条对角线互相垂直．········································· 6分
（3）线段．········································································································ 7分
点在抛物线，且，
得直线的解析式为，
解方程组，得点······················································· 8分
由对称性得点，··················································· 9分
，
．                                                      10分

解析