【答案】
分析:(1)將A、B、C的坐標(biāo)代入拋物線的解析式中,即可求得待定系數(shù)的值;
(2)根據(jù)(1)得到的拋物線的解析式,可求出其對(duì)稱軸方程聯(lián)立直線OD的解析式即可求出D點(diǎn)的坐標(biāo);由于⊙D與x軸相切,那么D點(diǎn)縱坐標(biāo)即為⊙D的半徑;欲求劣弧EF的長(zhǎng),關(guān)鍵是求出圓心角∠EDF的度數(shù),連接DE、DF,過D作y軸的垂線DM,則DM即為D點(diǎn)的橫坐標(biāo),通過解直角三角形易求得∠EDM和∠FDM的度數(shù),即可得到∠EDF的度數(shù),進(jìn)而可根據(jù)弧長(zhǎng)計(jì)算公式求出劣弧EF的長(zhǎng);
(3)易求得直線AC的解析式,設(shè)直線AC與PG的交點(diǎn)為N,設(shè)出P點(diǎn)的橫坐標(biāo),根據(jù)拋物線與直線AC的解析式即可得到P、N的縱坐標(biāo),進(jìn)而可求出PN,NG的長(zhǎng);Rt△PGA中,△PNA與△NGA同高不等底,那么它們的面積比等于底邊PN、NG的比,因此本題可分兩種情況討論:
①△PNA的面積是△NGA的2倍,則PN:NG=2:1;②△PNA的面積是△NGA的
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/0.png)
,則NG=2PN;
可根據(jù)上述兩種情況所得的不同等量關(guān)系求出P點(diǎn)的橫坐標(biāo),進(jìn)而由拋物線的解析式確定出P點(diǎn)的坐標(biāo).
解答:解:(1)∵拋物線y=ax
2+bx+c經(jīng)過點(diǎn)A(2,0),B(6,0),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/1.png)
;
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/2.png)
,
解得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/3.png)
;
∴拋物線的解析式為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/4.png)
;(3分)
(2)易知拋物線的對(duì)稱軸是x=4,
把x=4代入y=2x,得y=8,
∴點(diǎn)D的坐標(biāo)為(4,8);
∵⊙D與x軸相切,∴⊙D的半徑為8;(1分)
連接DE、DF,作DM⊥y軸,垂足為點(diǎn)M;
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/images5.png)
在Rt△MFD中,F(xiàn)D=8,MD=4,
∴cos∠MDF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/5.png)
;
∴∠MDF=60°,
∴∠EDF=120°;(2分)
∴劣弧EF的長(zhǎng)為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/6.png)
;(1分)
(3)設(shè)直線AC的解析式為y=kx+b;
∵直線AC經(jīng)過點(diǎn)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/7.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/8.png)
,
解得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/9.png)
;
∴直線AC的解析式為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/10.png)
;(1分)
設(shè)點(diǎn)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/11.png)
,PG交直線AC于N,
則點(diǎn)N坐標(biāo)為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/12.png)
,
∵S
△PNA:S
△GNA=PN:GN;
∴①若PN:GN=1:2,則PG:GN=3:2,PG=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/13.png)
GN;
即
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/14.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/15.png)
;
解得:m
1=-3,m
2=2(舍去);
當(dāng)m=-3時(shí),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/16.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/17.png)
;
∴此時(shí)點(diǎn)P的坐標(biāo)為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/18.png)
;(2分)
②若PN:GN=2:1,則PG:GN=3:1,PG=3GN;
即
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/19.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/20.png)
;
解得:m
1=-12,m
2=2(舍去);
當(dāng)m
1=-12時(shí),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/21.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/22.png)
;
∴此時(shí)點(diǎn)P的坐標(biāo)為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/23.png)
;
綜上所述,當(dāng)點(diǎn)P坐標(biāo)為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/24.png)
或
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101552935727457/SYS201311031015529357274026_DA/25.png)
時(shí),△PGA的面積被直線AC分成1:2兩部分.(2分)
點(diǎn)評(píng):此題主要考查了二次函數(shù)解析式的確定、函數(shù)圖象交點(diǎn)、圖形面積的求法等知識(shí),需要特別注意的是(3)題中,△PGA被直線AC所分成的兩部分中,并沒有明確誰大誰小,所以要分類討論,以免漏解.