【答案】
分析:(1)由題意,由于告訴了橢圓為焦點(diǎn)在x軸的橢圓所以可以利用定義設(shè)出 方程,然后建立a,b的方程求解即可;
(2)問是否存在的問題在圓錐曲線中就先假設(shè)存在,分斜率存在于不存在加以討論,并把直線方程與橢圓方程進(jìn)行連聯(lián)立,利用設(shè)而不求整體代換進(jìn)行求解.
解答:解:(Ⅰ)設(shè)橢圓C的方程為:
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/0.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/1.png)
=1(a>b>0),則a
2-b
2=1.①
∵當(dāng)l垂直于x軸時(shí),A,B兩點(diǎn)坐標(biāo)分別是(1,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/2.png)
)和(1,-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/3.png)
),
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/4.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/5.png)
=(1,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/6.png)
)•(1,-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/7.png)
)=1-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/8.png)
,則1-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/9.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/10.png)
,即a
2=6b
4.②
由①,②消去a,得6b
4-b
2-1=0.∴b
2=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/11.png)
或b
2=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/12.png)
.
當(dāng)b
2=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/13.png)
時(shí),a
2=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/14.png)
.因此,橢圓C的方程為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/15.png)
+2y
2=1.
(Ⅱ)設(shè)存在滿足條件的直線l.
(1)當(dāng)直線l垂直于x軸時(shí),由(Ⅰ)的解答可知|AB|=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/16.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/17.png)
,焦點(diǎn)F到右準(zhǔn)線的距離為d=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/18.png)
-c=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/19.png)
,
此時(shí)不滿足d=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/20.png)
|AB|.
因此,當(dāng)直線l垂直于x軸時(shí)不滿足條件.
(2)當(dāng)直線l不垂直于x軸時(shí),設(shè)直線l的斜率為k,則直線l的方程為y=k(x-1).
由
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/21.png)
⇒(6k
2+2)x
2-12k
2x+6k
2-3=0,
設(shè)A,B兩點(diǎn)的坐標(biāo)分別為(x
1,y
1)和(x
2,y
2),則x
1+x
2=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/22.png)
,x
1x
2=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/23.png)
.
|AB|=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/24.png)
|x
1-x
2|=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/25.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/26.png)
=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/27.png)
.
又設(shè)AB的中點(diǎn)為M,則x
M=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/28.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/29.png)
.
當(dāng)△ABP為正三角形時(shí),直線MP的斜率為k
MP=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/30.png)
.
∵x
p=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/31.png)
,∴|MP|=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/32.png)
|x
p-x
M|=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/33.png)
•(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/34.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/35.png)
)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/36.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/37.png)
.
當(dāng)△ABP為正三角形時(shí),|MP|=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/38.png)
|AB|,即
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/39.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/40.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/41.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768374522/SYS201310241809287683745003_DA/42.png)
,
解得k
2=1,k=±1.
因此,滿足條件的直線l存在,且直線l的方程為x-y-1=0或x+y-1=0.
點(diǎn)評:(1)次問重點(diǎn)考查了利用方程的思想由題意列出變量a,b的兩個(gè)方程,然后求解曲線的軌跡方程;
(2)次問重點(diǎn)考查了分類討論的思想及把直線方程與圓錐曲線方程進(jìn)行聯(lián)立設(shè)而不求整體代換的思想,還有對于圓錐曲線中是否存在利用假設(shè)的解題方法.