已知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401443522.png)
是橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240224014581170.png)
的右焦點(diǎn),圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401474802.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401474266.png)
軸交于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401489436.png)
兩點(diǎn),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401505309.png)
是橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401536313.png)
與圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401552302.png)
的一個(gè)交點(diǎn),且
(Ⅰ)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401536313.png)
的離心率;
(Ⅱ)過點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401505309.png)
與圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401552302.png)
相切的直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401630280.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401536313.png)
的另一交點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401661300.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401677495.png)
的面積為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401708694.png)
,求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401536313.png)
的方程
(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401723439.png)
;(Ⅱ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401739817.png)
試題分析:(Ⅰ)根據(jù)橢圓的定義、幾何性質(zhì)可求;(Ⅱ)直線與橢圓相交,聯(lián)立消元,設(shè)點(diǎn)代入化簡(jiǎn),利用點(diǎn)到直線的距離來求
試題解析:(Ⅰ)由題意,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401755555.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401817616.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401833610.png)
,
∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401567713.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401864669.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401942823.png)
,
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401973922.png)
, 得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401973429.png)
,
即橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401536313.png)
的離心率
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401723439.png)
(4分)
(Ⅱ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401536313.png)
的離心率
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401723439.png)
,令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401973429.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402082495.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402098912.png)
直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401630280.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402129416.png)
,設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402145867.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240224021601386.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402176955.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402191834.png)
又點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402207579.png)
到直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401630280.png)
的距離
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402238858.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402254527.png)
的面積
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402269812.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240224022851182.png)
, 解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022402301412.png)
故橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022401739817.png)
(12分)
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖,已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353176306.png)
:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240303531921085.png)
的離心率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353223452.png)
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的左頂點(diǎn)
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為圓心作圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353254304.png)
:
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,設(shè)圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353254304.png)
與橢圓
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353317399.png)
與點(diǎn)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240303533485457.png)
(1)求橢圓
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353379645.png)
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的方程;(4分)
(3)設(shè)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353395289.png)
是橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353176306.png)
上異于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353317399.png)
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,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353488291.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030353504543.png)
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的兩個(gè)端點(diǎn)在拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022534778463.png)
上滑動(dòng),則線段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022534762396.png)
中點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022534809378.png)
到
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過點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915446517.png)
作曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915462313.png)
:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915478454.png)
的切線,切點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915509319.png)
,設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915509319.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915540266.png)
軸上的投影是點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915556327.png)
,過點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915556327.png)
再作曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915462313.png)
的切線,切點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915602365.png)
,設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915602365.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915540266.png)
軸上的投影是點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915649361.png)
,…,依次下去,得到第
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915665365.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915680567.png)
個(gè)切點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915696411.png)
.則點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015915696411.png)
的坐標(biāo)為
.
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