問題詳情:
如圖14,已知半徑為1的與軸交於兩點,為的切線,切點為,圓心的座標為,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座標;若不存在,請説明理由.
【回答】
解:(1)圓心的座標為,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,(圓的切線垂直於經過切點的半徑).
在中,
為鋭角,···························· 5分
,
在中,.
.
點座標為························································································· 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,與交於點.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
符合條件的點座標有,······················································ 12分
知識點:相似三角形
題型:綜合題