如圖16,在平面直角坐標(biāo)系中,直線與
軸交于點
,與
軸交于點
,拋物線
經(jīng)過
三點.
(1)求過三點拋物線的解析式并求出頂點
的坐標(biāo);
(2)在拋物線上是否存在點,使
為直角三角形,若存在,直接寫出
點坐標(biāo);若不存在,請說明理由;
(3)試探究在直線上是否存在一點
,使得
的周長最小,若存在,求出
點的坐標(biāo);若不存在,請說明理由.
解:(1)直線
與
軸交于點
,與
軸交于點
.
,
························································································· 1分
點
都在拋物線上,
拋物線的解析式為
························································ 3分
頂點
······························································································· 4分
(2)存在··············································································································· 5分
············································································································· 7分
············································································································ 9分
(3)存在·············································································································· 10分
理由:
解法一:
延長到點
,使
,連接
交直線
于點
,則點
就是所求的點.
····················································································· 11分
過點
作
于點
.
點在拋物線
上,
在中,
,
,
,
在中,
,
,
,
··············································· 12分
設(shè)直線的解析式為
解得
································································································ 13分
解得
在直線
上存在點
,使得
的周長最小,此時
.··· 14分
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