Question

如圖14，已知半徑為1的與軸交于兩點，為的切線，切點為，圓心的坐標為，二次函數的圖象經過兩點．

（1）求二次函數的解析式；

（2）求切線的函數解析式；

（3）線段上是否存在一點，使得以為頂點的三角形與相似．若存在，請求出所有符合條件的點的坐標；若不存在，請說明理由．

Answer 1

解：（1）圓心的坐標為，半徑為1，，……1分

二次函數的圖象經過點，

可得方程組················································································ 2分

解得：二次函數解析式為············································· 3分

（2）過點作軸，垂足為．······························································· 4分

是的切線，為切點，（圓的切線垂直于經過切點的半徑）．

在中，

為銳角，···························· 5分

，

在中，．

．

點坐標為························································································· 6分

設切線的函數解析式為，由題意可知，······ 7分

切線的函數解析式為···································································· 8分

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

①過點作軸，與交于點．可得（兩角對應相等兩三角形相似）

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

②過點作，垂足為，過點作，垂足為．

可得（兩角對應相等兩三角開相似）

在中，，，

在中，，

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

符合條件的點坐標有，······················································ 12分