【答案】
分析:分類討論:當(dāng)k=0,方程變形為:x-1=0,解得x=1;當(dāng)k≠0,先運(yùn)用因式分解法解一元二次方程得到x
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/0.png)
,x
2=1,由于關(guān)于x的方程kx
2-(2k-1)x+k-1=0(k為整數(shù))的根為整數(shù),則由x
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/1.png)
=1-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/2.png)
得到k=1,得到雙曲線的解析式為y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/3.png)
,設(shè)M點(diǎn)坐標(biāo)為(a,b),易得C點(diǎn)坐標(biāo)為(a,0),B點(diǎn)坐標(biāo)為(a,2b),則A點(diǎn)的縱坐標(biāo)為2b,當(dāng)點(diǎn)A在雙曲線y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/4.png)
上,所以A點(diǎn)坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/5.png)
,2b),然后根據(jù)梯形面積公式進(jìn)行計(jì)算,四邊形OABC的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/6.png)
(AB+OC)•BC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/7.png)
(a-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/8.png)
+a)×2b=2ab-1,ab=2,可計(jì)算出面積;當(dāng)點(diǎn)A在雙曲線y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/9.png)
上,則A點(diǎn)坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/10.png)
,2b),四邊形OABC的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/11.png)
(AB+OC)•BC
=2ab-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/12.png)
,然后把a(bǔ)b=1代入計(jì)算.
解答:解:當(dāng)k=0,方程變形為:x-1=0,解得x=1;
當(dāng)k≠0,
∵[kx-(k-1)](x-1)=0,
∴x
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/13.png)
,x
2=1,
∵關(guān)于x的方程kx
2-(2k-1)x+k-1=0(k為整數(shù))的根為整數(shù),
而x
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/14.png)
=1-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/15.png)
,
∴k=1,
∴雙曲線的解析式為y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/16.png)
或y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/17.png)
,
設(shè)M點(diǎn)坐標(biāo)為(a,b),
∵四邊形OACB為梯形,∠BCO=90°,且M為BC的中點(diǎn),
∴C點(diǎn)坐標(biāo)為(a,0),B點(diǎn)坐標(biāo)為(a,2b),
∴A點(diǎn)的縱坐標(biāo)為2b,
當(dāng)點(diǎn)A在雙曲線y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/18.png)
上,
∴當(dāng)y=2b時(shí),x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/19.png)
,
∴A點(diǎn)坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/20.png)
,2b),
∴四邊形OACB的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/21.png)
(AB+OC)•BC
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/22.png)
(a-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/23.png)
+a)×2b
=2ab-1
當(dāng)k=1,ab=2,四邊形OACB的面積=4-1=3;
當(dāng)點(diǎn)A在雙曲線y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/24.png)
上,
A點(diǎn)坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/25.png)
,2b),
∴四邊形OACB的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/26.png)
(AB+OC)•BC
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/27.png)
(a-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/28.png)
+a)×2b
=2ab-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/29.png)
當(dāng)k=0,ab=1,四邊形OACB的面積=2-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/30.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/31.png)
;
∴四邊形OABC的面積為3或
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101192233800034319/SYS201311011922338000343019_DA/32.png)
.
點(diǎn)評(píng):本題考查了反比例函數(shù)綜合題:點(diǎn)在反比例函數(shù)圖象上,點(diǎn)的橫縱坐標(biāo)滿足圖象的解析式;平行于x軸的直線上所有點(diǎn)的縱坐標(biāo)相同;平行于y軸的直線上所有點(diǎn)的橫坐標(biāo)相同;運(yùn)用梯形的面積確定線段平行關(guān)系和計(jì)算面積;運(yùn)用因式分解法解一元二次方程;分類討論的思想方法在解題常用到.