問題詳情:
如圖14,已知半徑爲1的與軸交於兩點,爲的切線,切點爲,圓心的座標爲,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,使得以爲頂點的三角形與相似.若存在,請求出所有符合條件的點的座標;若不存在,請說明理由.
【回答】
解:(1)圓心的座標爲,半徑爲1,,……1分
二次函數的圖象經過點,
可得方程組················································································ 2分
解得:二次函數解析式爲············································· 3分
(2)過點作軸,垂足爲.······························································· 4分
是的切線,爲切點,(圓的切線垂直於經過切點的半徑).
在中,
爲銳角,···························· 5分
,
在中,.
.
點座標爲························································································· 6分
設切線的函數解析式爲,由題意可知,······ 7分
切線的函數解析式爲···································································· 8分
(3)存在.············································································································ 9分
①過點作軸,與交於點.可得(兩角對應相等兩三角形相似)
,··········································· 10分
②過點作,垂足爲,過點作,垂足爲.
可得(兩角對應相等兩三角開相似)
在中,,,
在中,,
,······································ 11分
符合條件的點座標有,······················································ 12分
知識點:相似三角形
題型:綜合題