Question

如圖14，已知半徑為1的與軸交于兩點(diǎn)，為的切線，切點(diǎn)為，圓心的坐標(biāo)為，二次函數(shù)的圖象經(jīng)過兩點(diǎn)．

（1）求二次函數(shù)的解析式；

（2）求切線的函數(shù)解析式；

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

Answer 1

解：（1）圓心的坐標(biāo)為，半徑為1，，……1分

二次函數(shù)的圖象經(jīng)過點(diǎn)，

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

解得：二次函數(shù)解析式為············································· 3分

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

是的切線，為切點(diǎn)，（圓的切線垂直于經(jīng)過切點(diǎn)的半徑）．

在中，

為銳角，···························· 5分

，

在中，．

．

點(diǎn)坐標(biāo)為························································································· 6分

設(shè)切線的函數(shù)解析式為，由題意可知，······ 7分

切線的函數(shù)解析式為···································································· 8分

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

①過點(diǎn)作軸，與交于點(diǎn)．可得（兩角對應(yīng)相等兩三角形相似）

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

②過點(diǎn)作，垂足為，過點(diǎn)作，垂足為．

可得（兩角對應(yīng)相等兩三角開相似）

在中，，，

在中，，

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

符合條件的點(diǎn)坐標(biāo)有，······················································ 12分