【答案】
分析:(1)由根與系數(shù)的關(guān)系可得到a+h及ah的值,然后分別表示出正方形和矩形的面積,再根據(jù)根的判別式進(jìn)行判斷即可;
(2)過(guò)D、E、F三點(diǎn)的⊙O一定是以DF為直徑的圓,那么其面積為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/0.png)
(EF
2+DE
2);而矩形PDEF的面積為:EF•DE;那么
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/1.png)
,可將
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/2.png)
看作一個(gè)整體,將兩個(gè)圖形的面積比轉(zhuǎn)化為完全平方式,進(jìn)而得出其最小值;
(3)過(guò)B作BM⊥AQ于M,交直線PF于N;易證得△FBP∽△ABQ,根據(jù)相似三角形的對(duì)應(yīng)線段成比例可得EP:AQ=BN:BM;而當(dāng)(2)的面積比最小時(shí),EF=DE,此時(shí)BN=FP,即AQ=BM=h;h是已知方程的一個(gè)根,由此可判斷出AQ的長(zhǎng)是否與m、n、k的取值有關(guān).
解答:解:解法一:
(1)據(jù)題意,∵a+h=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/3.png)
,ah=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/4.png)
∴所求正方形與矩形的面積之比:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/5.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/6.png)
(1分)
∵n
2-4mk≥0,∴n
2≥4mk,由
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/7.png)
知m,k同號(hào),
∴mk>0 (2分)
(說(shuō)明:此處未得出mk>0只扣(1分),不再影響下面評(píng)分)
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/8.png)
(3分)
即正方形與矩形的面積之比不小于4.
(2)∵∠FED=90°,∴DF為⊙O的直徑.
∴⊙O的面積為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/9.png)
. (4分)
矩形PDEF的面積:S
矩形PDEF=EF•DE.
∴面積之比:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/10.png)
,設(shè)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/11.png)
.
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/12.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/13.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/14.png)
.(6分)
∵
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/15.png)
,∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/16.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/17.png)
,即f=1時(shí)(EF=DE),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/18.png)
的最小值為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/19.png)
(7分)
(3)當(dāng)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/20.png)
的值最小時(shí),這時(shí)矩形PDEF的四邊相等為正方形.
過(guò)B點(diǎn)過(guò)BM⊥AQ,M為垂足,BM交直線PF于N點(diǎn),設(shè)FP=e,
∵BN∥FE,NF∥BE,∴BN=EF,∴BN=FP=e.
由BC∥MQ,得:BM=AG=h.
∵AQ∥BC,PF∥BC,∴AQ∥FP,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/images21.png)
∴△FBP∽△ABQ. (8分)
(說(shuō)明:此處有多種相似關(guān)系可用,要同等分步驟評(píng)分)
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/21.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/22.png)
,(9分)
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/23.png)
,∴AQ=h (10分)
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/24.png)
(11分)
∴線段AQ的長(zhǎng)與m,n,k的取值有關(guān).
(解題過(guò)程敘述基本清楚即可)
解法二:
(1)∵a,h為線段長(zhǎng),即a,h都大于0,
∴ah>0 (1分)(說(shuō)明:此處未得出ah>0只扣(1分),再不影響下面評(píng)分)
∵(a-h)
2≥0,當(dāng)a=h時(shí)等號(hào)成立.
故,(a-h)
2=(a+h)
2-4ah≥0.(2分)
∴(a+h)
2≥4ah,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/25.png)
≥4.(﹡) (3分)
這就證得
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/26.png)
≥4.(敘述基本明晰即可)
(2)設(shè)矩形PDEF的邊PD=x,DE=y,則⊙O的直徑為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/27.png)
.
S
⊙O=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/28.png)
(4分),S
矩形PDEF=xy
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/29.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/30.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/31.png)
(6分)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/32.png)
由(1)(*).
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/33.png)
.
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/34.png)
的最小值是
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/35.png)
(7分)
(3)當(dāng)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/36.png)
的值最小時(shí),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/images38.png)
這時(shí)矩形PDEF的四邊相等為正方形.
∴EF=PF.作AG⊥BC,G為垂足.
∵△AGB∽△FEB,∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/37.png)
. (8分)
∵△AQB∽△FPB,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/38.png)
,(9分)
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/39.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/40.png)
.
而EF=PF,∴AG=AQ=h,(10分)
∴AG=h=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/41.png)
,
或者AG=h=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101719103423856/SYS201311031017191034238011_DA/42.png)
(11分)
∴線段AQ的長(zhǎng)與m,n,k的取值有關(guān).
(解題過(guò)程敘述基本清楚即可)
點(diǎn)評(píng):此題主要考查了矩形的性質(zhì)、相似三角形的判定和性質(zhì)以及二次函數(shù)的應(yīng)用等知識(shí),綜合性強(qiáng),難度較大.