【答案】
分析:(1)將A、C的坐標(biāo)代入拋物線中進(jìn)行求解即可.
(2)先根據(jù)拋物線的解析式求出B點(diǎn)的坐標(biāo),即可求出OB的長(zhǎng),然后設(shè)M的坐標(biāo)為(m,0),可用m表示OM和NC的長(zhǎng),然后根據(jù)三角形的面積公式即可得出關(guān)于△CMN的面積與m之間的函數(shù)關(guān)系式,根據(jù)函數(shù)的性質(zhì)和m的取值范圍即可求出△CNM的最大值及對(duì)應(yīng)的M、N的坐標(biāo).
(3)本題要分三種情況進(jìn)行討論:
①OF=DF,此時(shí)F必在OD的垂直平分線上,即F的橫坐標(biāo)為-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/0.png)
,可根據(jù)直線AC的解析式求出F點(diǎn)的坐標(biāo),然后將F的縱坐標(biāo)代入拋物線中即可求出P點(diǎn)的坐標(biāo).
②OD=DF,DF=1,易知:OA=OC=2,因此AD=OD=DF=1,三角形AFO為等腰直角三角形,因此可得出F(-1,1),后同①.
③當(dāng)OD=DF=1時(shí),②中已經(jīng)得出△OAC為等腰直角三角形,因此O到AC的距離為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/1.png)
>1,因此這種情況不成立.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/images2.png)
解:(1)由題意,得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/2.png)
解得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/3.png)
∴所求拋物線的解析式為:y=-x
2-x+2
(2)設(shè)點(diǎn)M的坐標(biāo)為(m,0),則OM=m,ON=2m,CN=2-2m.
S
△MNC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/4.png)
NC•OM=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/5.png)
(2-2m)•m=-m
2+m=-(m-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/6.png)
)
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/7.png)
由-x
2-x+2=0
得x
1=-2,x
2=1.
∴點(diǎn)B的坐標(biāo)為(1,0).
則0<m<1,
∴當(dāng)m=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/8.png)
時(shí),S
△MNC有最大值
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/9.png)
此時(shí),點(diǎn)M的坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/10.png)
,0),點(diǎn)N的坐標(biāo)為(0,1).
(3)在△ODF中,
①若DO=DF,∵A(-2,0),D(-1,0),
∴AD=DO=DF=1.
又在Rt△AOC中,OA=OC=2,
∴∠OAC=45度.
∴∠DFA=∠OAC=45度.
∴∠ADF=90度.此時(shí),點(diǎn)F的坐標(biāo)為(-1,1).
由-x
2-x+2=1,x;
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/11.png)
,x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/12.png)
此時(shí),
點(diǎn)P的坐標(biāo)為:(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/13.png)
,1)或(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/14.png)
,1)
②若FO=FD,過(guò)點(diǎn)F作FE⊥x軸于點(diǎn)E.
由等腰三角形△AEF中,F(xiàn)E=AE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/15.png)
.
∴F(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/16.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/17.png)
)
由-x
2-x+2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/18.png)
,
得x
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/19.png)
,x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/20.png)
此時(shí),
點(diǎn)P的坐標(biāo)為:(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/21.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/22.png)
)或(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/23.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/24.png)
)
③若OF=OD,∵OA=OC=2,且∠AOC=90°,
∴AC=2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/25.png)
.
∴點(diǎn)O到AC的距離為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/26.png)
,而OF=OD=1<
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/27.png)
,
此時(shí),不存在這樣的直線l,使得△ODF是等腰三角形.
綜上所述,存在這樣的直線l,使得△ODF是等腰三角形,
所求點(diǎn)P的坐標(biāo)為:(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/28.png)
,1)或(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/29.png)
,1)或(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/30.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/31.png)
)或(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/32.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/201310201208069651837054/SYS201310201208069651837022_DA/33.png)
).
點(diǎn)評(píng):本題考查了二次函數(shù)解析式的確定、函數(shù)圖象交點(diǎn)、等腰三角形的判定等知識(shí)點(diǎn),綜合性強(qiáng),考查學(xué)生分類討論、數(shù)形結(jié)合的數(shù)學(xué)思想方法.