試題分析:本題考查橢圓的定義、余弦定理及韋達(dá)定理的應(yīng)用.第一問是利用三角形面積公式、余弦定理、橢圓的定義,三個(gè)方程聯(lián)立,解出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021920845283.png)
,再根據(jù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021920860450.png)
的關(guān)系求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021920876299.png)
,本問分析已知條件是解題的關(guān)鍵;第二問是直線與橢圓相交于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021920767423.png)
兩點(diǎn),先設(shè)出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021920767423.png)
兩點(diǎn)坐標(biāo),本題的突破口是在消參后的方程中找出兩根之和、兩根之積,整理斜率的表達(dá)式,但是在本問中需考慮直線的斜率是否存在,此題中蘊(yùn)含了分類討論的思想的應(yīng)用.
試題解析:(Ⅰ)在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021920642657.png)
中,
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240219210011348.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021921016891.png)
.
由余弦定理,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240219210321463.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240219210481614.png)
,
從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240219210631019.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021921079478.png)
,從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021921110403.png)
,
故橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021920502313.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021920829710.png)
. 6分
(Ⅱ)當(dāng)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021920720280.png)
的斜率存在時(shí),設(shè)其方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021921172717.png)
,
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240219211881258.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240219212041130.png)
. 8分
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021921219616.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021921250644.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240219212661005.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021921282920.png)
.
從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240219212972944.png)
. 11分
當(dāng)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021920720280.png)
的斜率不存在時(shí),得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240219213281028.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021921344542.png)
.
綜上,恒有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021921344542.png)
. 12分