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).当时，求点的坐标；

 

 (2).当为何值时，四边形的两条对角线互相垂直；

(3).猜想线段与之间的数量关系，并证明你的结论．

 

Answer 1

【答案】

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】略