Question

如图14，已知半径为1的与轴交于两点，为的切线，切点为，圆心的坐标为，二次函数的图象经过两点．

（1）求二次函数的解析式；

（2）求切线的函数解析式；

（3）线段上是否存在一点，使得以为顶点的三角形与相似．若存在，请求出所有符合条件的点的坐标；若不存在，请说明理由．

Answer 1

解：（1）圆心的坐标为，半径为1，，……1分

二次函数的图象经过点，

可得方程组················································································ 2分

解得：二次函数解析式为············································· 3分

（2）过点作轴，垂足为．······························································· 4分

是的切线，为切点，（圆的切线垂直于经过切点的半径）．

在中，

为锐角，···························· 5分

，

在中，．

．

点坐标为························································································· 6分

设切线的函数解析式为，由题意可知，······ 7分

切线的函数解析式为···································································· 8分

（3）存在．············································································································ 9分

①过点作轴，与交于点．可得（两角对应相等两三角形相似）

，··········································· 10分

②过点作，垂足为，过点作，垂足为．

可得（两角对应相等两三角开相似）

在中，，，

在中，，

，······································ 11分

符合条件的点坐标有，······················································ 12分