在平面直角坐標系
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557264413.png)
中,橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557264313.png)
的標準方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240155572791112.png)
,右焦點為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557295302.png)
,右準線為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557310280.png)
,短軸的一個端點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557326309.png)
. 設原點到直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557342393.png)
的距離為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557342342.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557295302.png)
點到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557310280.png)
的距離為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557388388.png)
. 若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557404591.png)
,則橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557264313.png)
的離心率為
依題意,作
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557451568.png)
于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557264313.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557466565.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557498737.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557544603.png)
,而橢圓準線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557310280.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557591525.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557607493.png)
,設直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557310280.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557638266.png)
軸交于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557654300.png)
,則點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557295302.png)
到直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557310280.png)
的距離
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557700883.png)
,∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557404591.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557732788.png)
,整理的
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557732593.png)
,兩邊平方,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557747703.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557763597.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557778422.png)
,
解
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557763597.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557810520.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240155578104544.jpg)
【考點定位】橢圓的性質、點到直線的距離公式,考查分析轉化能力、計算能力.中等題.
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941350310.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941365634.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546889313.png)
的兩個焦點為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546904719.png)
,點
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在橢圓
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上.
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546889313.png)
的方程;
(Ⅱ)已知點
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,設點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546998289.png)
是橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546889313.png)
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科目:高中數(shù)學
來源:不詳
題型:解答題
(本題滿分14分)
已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240020459841165.png)
過點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046000534.png)
,且離心率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046016413.png)
.
(1)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046031313.png)
的方程;
(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046047423.png)
為橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046031313.png)
的左右頂點,點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046094289.png)
是橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046031313.png)
上異于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046047423.png)
的動點,直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046218491.png)
分別交直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046250551.png)
于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046250426.png)
兩點.
證明:以線段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046265386.png)
為直徑的圓恒過
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002046296266.png)
軸上的定點.
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