【答案】
分析:(1)已知了拋物線的解析式,不難用公式法求出M的坐標(biāo)為(1,a-1).由于拋物線過(guò)A點(diǎn),因此A的坐標(biāo)是(0,a).根據(jù)A,M的坐標(biāo),用待定系數(shù)法可得出直線AM的解析式為y=-x+a.直線AM和y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/0.png)
x-a聯(lián)立方程組即可求出N的坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/1.png)
a,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/2.png)
a).
(2)根據(jù)折疊的性質(zhì)不難得出N與N′正好關(guān)于y軸對(duì)稱,因此N′的坐標(biāo)為(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/3.png)
a,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/4.png)
a).由于N′在拋物線上,因此將N′的坐標(biāo)代入拋物線的解析式中即可得出a的值.也就能確定N,C的坐標(biāo).求四邊形ADCN的面積,可分成△ANC和△ADC兩部分來(lái)求.已經(jīng)求得了A,C,N的坐標(biāo),可求出AC的長(zhǎng)以及N,D到y(tǒng)軸的距離.也就能求出△ANC和△ADC的面積,進(jìn)而可求出四邊形ADCN的面積.
(3)本題可分兩種情況進(jìn)行討論:
①當(dāng)P在y軸左側(cè)時(shí),如果使以P,N,A,C為頂點(diǎn)的四邊形為平行四邊形,那么P需要滿足的條件是PN平行且相等于AC,也就是說(shuō),如果N點(diǎn)向上平移AC個(gè)單位即-2a后得到的點(diǎn)就是P點(diǎn).然后將此時(shí)P的坐標(biāo)代入拋物線中,如果沒有解說(shuō)明不存在這樣的點(diǎn)P,如果能求出a的值,那么即可求出此時(shí)P的坐標(biāo).
②當(dāng)P在y軸右側(cè)時(shí),P需要滿足的條件是PN與AC應(yīng)互相平分(平行四邊形的對(duì)角線互相平分),那么NP必過(guò)原點(diǎn),且關(guān)于原點(diǎn)對(duì)稱.那么可得出此時(shí)P的坐標(biāo),然后代入拋物線的解析式中按①的方法求解即可.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/images5.png)
解:(1)M(1,a-1),N(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/5.png)
a,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/6.png)
a);
(2)∵由題意得點(diǎn)N與點(diǎn)N′關(guān)于y軸對(duì)稱,
∴N′(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/7.png)
a,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/8.png)
a).
將N′的坐標(biāo)代入y=x
2-2x+a得:
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/9.png)
a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/10.png)
a
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/11.png)
a+a,
∴a
1=0(不合題意,舍去),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/12.png)
.
∴N(-3,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/13.png)
),
∴點(diǎn)N到y(tǒng)軸的距離為3.
∵A(0,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/14.png)
),N'(3,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/15.png)
),
∴直線AN'的解析式為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/16.png)
,它與x軸的交點(diǎn)為D(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/17.png)
)
∴點(diǎn)D到y(tǒng)軸的距離為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/18.png)
.
∴S
四邊形ADCN=S
△ACN+S
△ACD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/19.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/20.png)
×3+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/21.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/22.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/23.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/24.png)
;
(3)存在,理由如下:
當(dāng)點(diǎn)P在y軸的左側(cè)時(shí),若ACPN是平行四邊形,則PN平行且等于AC,
則把N向上平移-2a個(gè)單位得到P,坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/25.png)
a,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/26.png)
a),代入拋物線的解析式,
得:-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/27.png)
a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/28.png)
a
2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/29.png)
a+a,
解得a
1=0(不舍題意,舍去),a
2=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/30.png)
,
則P(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/31.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/32.png)
);
當(dāng)點(diǎn)P在y軸的右側(cè)時(shí),若APCN是平行四邊形,則AC與PN互相平分,
則OA=OC,OP=ON.
則P與N關(guān)于原點(diǎn)對(duì)稱,
則P(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/33.png)
a,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/34.png)
a);
將P點(diǎn)坐標(biāo)代入拋物線解析式得:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/35.png)
a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/36.png)
a
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/37.png)
a+a,
解得a
1=0(不合題意,舍去),a
2=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/38.png)
,
則P(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/39.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/40.png)
).
故存在這樣的點(diǎn)P
1(-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/41.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/42.png)
)或P
2(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/43.png)
,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101129573817237/SYS201311031011295738172025_DA/44.png)
),能使得以P,A,C,N為頂點(diǎn)的四邊形是平行四邊形.
點(diǎn)評(píng):本題著重考查了待定系數(shù)法求函數(shù)解析式、圖形旋轉(zhuǎn)變換、平行四邊形的性質(zhì)等重要知識(shí)點(diǎn),綜合性強(qiáng),能力要求較高.考查學(xué)生分類討論,數(shù)形結(jié)合的數(shù)學(xué)思想方法.