如圖,已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240449084421175.png)
的左、右焦點分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908457441.png)
,其上頂點為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908473310.png)
已知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908489578.png)
是邊長為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908504291.png)
的正三角形.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240449085203640.jpg)
(1)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908535313.png)
的方程;
(2)過點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908551574.png)
任作一動直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908567280.png)
交橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908535313.png)
于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908598550.png)
兩點,記
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908613795.png)
.若在線段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908629513.png)
上取一點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908645303.png)
,使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908676771.png)
,當(dāng)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908567280.png)
運動時,點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908645303.png)
在某一定直線上運動,求出該定直線的方程.
試題分析:(1)因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908489578.png)
是邊長為2的正三角形,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908769718.png)
,橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908535313.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908723707.png)
;(2)設(shè)直線方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908801648.png)
,與橢圓方程聯(lián)立,結(jié)合韋達定理,表示出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908816720.png)
;
設(shè)點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908645303.png)
的坐標(biāo)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908847576.png)
則由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908676771.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908879399.png)
,故點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908645303.png)
在定直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908738332.png)
上.
試題解析:(1)因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908489578.png)
是邊長為2的正三角形,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908769718.png)
,所以,橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908535313.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908723707.png)
(2)由題意知,直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908629513.png)
的斜率必存在,設(shè)其方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908801648.png)
.并設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044909019992.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240449090501179.png)
消去
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044909066310.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240449090971141.png)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044909113848.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044909128938.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908613795.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044909191742.png)
故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908816720.png)
設(shè)點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908645303.png)
的坐標(biāo)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908847576.png)
則由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908676771.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044909284737.png)
解得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240449093003630.png)
故點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908645303.png)
在定直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044908738332.png)
上.
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044116744215.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044116744215.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044116728423.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044116853377.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044116884315.png)
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