摘要：26．如图16.在平面直角坐标系中.直线与轴交于点.与轴交于点.抛物线经过三点． (1)求过三点抛物线的解析式并求出顶点的坐标, (2)在抛物线上是否存在点.使为直角三角形.若存在.直接写出点坐标,若不存在.请说明理由, (3)试探究在直线上是否存在一点.使得的周长最小.若存在.求出点的坐标,若不存在.请说明理由． 解:(1)直线与轴交于点.与轴交于点． .························································································· 1分 点都在抛物线上. 抛物线的解析式为························································ 3分 顶点······························································································· 4分 (2)存在··············································································································· 5分 ············································································································· 7分 ············································································································ 9分 (3)存在·············································································································· 10分 理由: 解法一: 延长到点.使.连接交直线于点.则点就是所求的点． ····················································································· 11分 过点作于点． 点在抛物线上. 在中.. .. 在中.. ..··············································· 12分 设直线的解析式为 解得 ································································································ 13分 解得 在直线上存在点.使得的周长最小.此时．··· 14分 解法二: 过点作的垂线交轴于点.则点为点关于直线的对称点．连接交于点.则点即为所求．················································································ 11分 过点作轴于点.则.． . 同方法一可求得． 在中...可求得. 为线段的垂直平分线.可证得为等边三角形. 垂直平分． 即点为点关于的对称点．············································· 12分 设直线的解析式为.由题意得 解得 ································································································ 13分 解得 在直线上存在点.使得的周长最小.此时．··· 14分

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