在平面直角坐標(biāo)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(biāo)準(zhǔn)方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240155572791112.png)
,右焦點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557295302.png)
,右準(zhǔn)線為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557310280.png)
,短軸的一個(gè)端點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015557326309.png)
. 設(shè)原點(diǎn)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(diǎn)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(zhǔn)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(shè)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(diǎn)
![](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(diǎn)定位】橢圓的性質(zhì)、點(diǎn)到直線的距離公式,考查分析轉(zhuǎn)化能力、計(jì)算能力.中等題.
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952185303.png)
:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240219522011105.png)
的長(zhǎng)軸長(zhǎng)為4,且過點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952217732.png)
.
(1)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952185303.png)
的方程;
(2)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952248300.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952263309.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952279399.png)
是橢圓上的三點(diǎn),若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952295959.png)
,點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952310357.png)
為線段
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952341396.png)
的中點(diǎn),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952357313.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952373315.png)
兩點(diǎn)的坐標(biāo)分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952388837.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952404818.png)
,求證:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952419837.png)
.
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖,在平面直角坐標(biāo)系
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651466449.png)
中,橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651498993.png)
的右焦點(diǎn)為
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分別過
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,
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的兩條弦
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,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651576392.png)
相交于點(diǎn)
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(異于
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,
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兩點(diǎn)),且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651638507.png)
.
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,
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的斜率之和為定值.
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來源:不詳
題型:解答題
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來源:不詳
題型:填空題
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015337862313.png)
:
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在圓
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015337955399.png)
、
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兩點(diǎn).
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015337862313.png)
的方程;
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824015338002663.png)
(
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知圓的方程為
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、
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,直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941022449.png)
恰好經(jīng)過橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240149410381086.png)
的右頂點(diǎn)和上頂點(diǎn).
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240149410533947.png)
(Ⅰ)求橢圓的方程;
(Ⅱ)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941069403.png)
是橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941085752.png)
(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941100565.png)
垂直于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941131271.png)
軸的一條弦,
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所在直線的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941163651.png)
且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941178598.png)
是橢圓上異于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941194302.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941209316.png)
的任意一點(diǎn),直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941225384.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941319375.png)
分別交定直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941334693.png)
于兩點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941350341.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941350310.png)
,求證
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014941365634.png)
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546889313.png)
的兩個(gè)焦點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546904719.png)
,點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546920676.png)
在橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546889313.png)
上.
(Ⅰ)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546889313.png)
的方程;
(Ⅱ)已知點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546967549.png)
,設(shè)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546998289.png)
是橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546889313.png)
上任一點(diǎn),求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013547029548.png)
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科目:高中數(shù)學(xué)
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(diǎn)
![](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(diǎn),點(diǎn)
![](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(dòng)點(diǎn),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(diǎn).
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(diǎn).
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