【答案】
分析:(1)由圖可知點(diǎn)M應(yīng)該是△ABC的重心,可依據(jù)平面直角坐標(biāo)系中,三角形重心的坐標(biāo)是三角形三頂點(diǎn)的算術(shù)平均數(shù)來求出重心M的坐標(biāo);
(2)可先根據(jù)A、B、C三點(diǎn)的坐標(biāo)和中位線定理求出A
1、B
1、C
1三點(diǎn)坐標(biāo),然后用待定系數(shù)法分別求出兩條拋物線的解析式;
(3)由于拋物線同時(shí)過E、F兩點(diǎn),可聯(lián)立(2)中兩個(gè)拋物線的解析式,然后得出一個(gè)關(guān)于x的一元二次方程,求出的兩個(gè)解便是E、F點(diǎn)的坐標(biāo).然后求出C
2的坐標(biāo),如果C2的橫坐標(biāo)大于或小于△EFC
1的所有頂點(diǎn)橫坐標(biāo),則說明C
2在△EFC1外,如果不是這樣,則E、F和C
1坐標(biāo)都知道了,則根據(jù)兩點(diǎn)式方程求出△EFC
1三邊所在直線的方程,將C2的橫坐標(biāo)分別代入這三個(gè)直線方程,
①如果求出的結(jié)果全大于或小于C
2的縱坐標(biāo),則說明C
2在△EFC
1外;
②如果求出的結(jié)果中有一至兩個(gè)等于C
2的縱坐標(biāo),則說明C
2在△EFC
1的邊上,甚至頂點(diǎn)上;
③如果求出的結(jié)果不全大于或小于C
2的縱坐標(biāo),則說明C
2在△EFC
1內(nèi).
(4)先用三點(diǎn)的坐標(biāo)確定一個(gè)拋物線的解析式,然后將剩下的一點(diǎn)代入拋物線中即可判斷出四點(diǎn)是否在同一拋物線上.
解答:解:(1)由題意可知:M點(diǎn)的坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/0.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/1.png)
),
即M(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/2.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/3.png)
);
(2)設(shè)過A、B、C三點(diǎn)的拋物線的解析式為y=a(x-0)(x-3),
則有:2=a×(2-0)×(2-3),解得a=-1,
因此過A、B、C三點(diǎn)的拋物線的解析式為y=-x
2+3x,
可求得A
1、B
1、C
1的坐標(biāo)分別為A
1(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/4.png)
,1),B
1(1,1),C
1(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/5.png)
,0),
設(shè)過這三點(diǎn)的拋物線的解析式為y=ax
2+bx+c,則有:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/6.png)
解得:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/7.png)
即A
1,B
1,C
1三點(diǎn)的拋物線解析式為y=2x
2-7x+6;
(3)根據(jù)題意有:2x
2-7x+6=-x
2+3x
即3x
2-10x+6=0
解得x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/8.png)
,x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/9.png)
由于E在F點(diǎn)左側(cè),
因此E(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/10.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/11.png)
),F(xiàn)(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/12.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/13.png)
)
由題意可知C
2的坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/14.png)
,1)
然后將C
2的坐標(biāo)代入△EFC
1三邊所在的直線中,可得出C
2在△EFC
1外;
(4)A,A
2,C,C
2四點(diǎn)的坐標(biāo)分別為:(0,0),(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/15.png)
,0)(2,2)(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/16.png)
,1),
設(shè)過A、A
2、C三點(diǎn)的拋物線的解析式為y=m(x-0)(x-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/17.png)
),
則有2=m×(2-0)×(2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/18.png)
),m=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/19.png)
,
因此拋物線的解析式為:y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/20.png)
x
2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/21.png)
x…①
將C
2點(diǎn)的坐標(biāo)代入①中可得:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/22.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/23.png)
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/24.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/25.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103103424901573458/SYS201311031034249015734014_DA/26.png)
≠1
因此:A,A
2,C,C
2四點(diǎn)不可能在同一條拋物線上.
點(diǎn)評(píng):本題著重考查了待定系數(shù)法求二次函數(shù)解析式、中位線定理、三角形重心等知識(shí)點(diǎn),綜合性強(qiáng).能力要求高.考查學(xué)生數(shù)形結(jié)合的數(shù)學(xué)思想方法.