【答案】
分析:(1)首先用c表示出M點(diǎn)的坐標(biāo),再由待定系數(shù)法可求出直線MC的表達(dá)式,已知點(diǎn)A(m,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/0.png)
-c)在直線MC上,且點(diǎn)B(-m,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/1.png)
-c)在拋物線上,通過(guò)聯(lián)立方程組即可求出m、c的值.
(2)由于四邊形的四頂點(diǎn)排序沒(méi)有明確,所以要分情況進(jìn)行討論,通過(guò)題意不難得出點(diǎn)C、N都在y軸上,所以:
①當(dāng)CN為平行四邊形的邊時(shí),那么AP與CN平行且相等,所以將點(diǎn)A向上或向下平移CN長(zhǎng)個(gè)單位即可得到點(diǎn)P的坐標(biāo)(有兩個(gè));
②當(dāng)CN為平行四邊形的對(duì)角線時(shí),由于平行四邊形是中心對(duì)稱圖形,且C、N關(guān)于原點(diǎn)對(duì)稱,所以點(diǎn)A、P必關(guān)于原點(diǎn)對(duì)稱,則P點(diǎn)坐標(biāo)可求.
解答:解:(1)由y=x
2-2x+c=(x-1)
2+c-1(c<0)知,M(1,c-1)、C(0,c);
設(shè)直線MC的解析式:y=kx+b,則有:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/2.png)
,
解得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/3.png)
故直線MC:y=-x+c;
∵A(m,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/4.png)
)是直線MC上的點(diǎn),
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/5.png)
-c=-m+c…①
∵點(diǎn)A關(guān)于y軸對(duì)稱點(diǎn)B(-m,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/6.png)
-c)在拋物線上,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/7.png)
-c=m
2-2(-m)+c…②
聯(lián)立①②,解得:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/8.png)
(舍),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/9.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/images10.png)
故拋物線對(duì)應(yīng)的函數(shù)式:y=x
2-2x-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/10.png)
.
(2)假設(shè)存在符合題意的平行四邊形;
由(1)知,A(-3,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/11.png)
)、C(0,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/12.png)
)、N(0,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/13.png)
);
①當(dāng)CN為平行四邊形的邊時(shí),CN
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/14.png)
AP,已知:CN=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/15.png)
,則有:
將點(diǎn)A向上平移
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/16.png)
個(gè)單位,得 P
1(-3,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/17.png)
);
將點(diǎn)A向下平移
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/18.png)
個(gè)單位,得 P
2(-3,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/19.png)
);
②當(dāng)CN為平行四邊形的對(duì)角線時(shí),點(diǎn)A、P關(guān)于原點(diǎn)對(duì)稱,
則 P
3(3,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/20.png)
);
又∵點(diǎn)P在拋物線上,
∴點(diǎn)P的坐標(biāo)為(3,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/21.png)
),
,綜上當(dāng)點(diǎn)P的坐標(biāo)為(3,-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192240606860768/SYS201311011922406068607025_DA/22.png)
)時(shí),以A、C、P、N為頂點(diǎn)的四邊形是平行四邊形.
點(diǎn)評(píng):此題主要考查了函數(shù)解析式的確定以及平行四邊形的判定和性質(zhì),在平行四邊形的四頂點(diǎn)排序不確定的情況下,一定要分類進(jìn)行討論.