已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354324301.png)
:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333543401094.png)
的左、右焦點分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354356642.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354371629.png)
,橢圓上的點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354402289.png)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354418694.png)
,且△
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354434486.png)
的面積為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354465795.png)
.
(Ⅰ)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354324301.png)
的方程;
(Ⅱ)設橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354324301.png)
的左、右頂點分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354512290.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354543299.png)
,過點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354558459.png)
的動直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354574269.png)
與橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354324301.png)
相交于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354605373.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354636333.png)
兩點,直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354652412.png)
與直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354683334.png)
的交點為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354699294.png)
,證明:點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354699294.png)
總在直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354761429.png)
上.
(Ⅰ)橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354777313.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354792657.png)
;(Ⅱ)詳見解析.
試題分析:(Ⅰ)由焦點坐標知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354808740.png)
.又橢圓上的點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354824296.png)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354855710.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354870818.png)
可求得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354902404.png)
,再由勾股定理可求得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354917431.png)
,從而求得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354933287.png)
.再由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354948603.png)
求得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354964294.png)
,從而得橢圓的方程.(Ⅱ)首先考慮
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354980275.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355011266.png)
軸垂直的情況,此時可求出直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355026452.png)
與直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355042360.png)
的交點為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355058637.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355089487.png)
的方程是:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355104831.png)
,代入驗證知點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355120310.png)
在直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355089487.png)
上.當直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354980275.png)
不與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355011266.png)
軸垂直時,設直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354980275.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355214631.png)
,點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355214718.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355245690.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355260645.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355276737.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355307900.png)
,要證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355323588.png)
共線,只需證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355338802.png)
,即證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333553701329.png)
.
若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355385394.png)
,顯然成立;若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355401413.png)
, 即證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333554161024.png)
而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333554321622.png)
,這顯然用韋達定理.
試題解析:(Ⅰ)由題意知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354808740.png)
, 1分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355463227.png)
橢圓上的點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354824296.png)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354855710.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354870818.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333555261894.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355541616.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333555571079.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355572946.png)
2分
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355588840.png)
3分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355604198.png)
橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354777313.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354792657.png)
. 4分
(Ⅱ)由題意知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355666582.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355682561.png)
,
(1)當直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354980275.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355011266.png)
軸垂直時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355744808.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355760784.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355026452.png)
的方程是:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355791875.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355089487.png)
的方程是:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355104831.png)
,直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355026452.png)
與直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355042360.png)
的交點為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355058637.png)
,
∴點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355120310.png)
在直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355089487.png)
上. 6分
(2)當直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354980275.png)
不與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355011266.png)
軸垂直時,設直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033354980275.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355214631.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355214718.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355245690.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355260645.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333560561086.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033356072975.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033356087875.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033356103829.png)
7分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033356134738.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033356150884.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033356165553.png)
共線,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033356196802.png)
8分
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355276737.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355307900.png)
,需證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355323588.png)
共線,
需證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355338802.png)
,只需證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333553701329.png)
若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355385394.png)
,顯然成立,若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355401413.png)
, 即證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333563211037.png)
∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333563371600.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240333563521287.png)
成立, 11分
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355323588.png)
共線,即點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355120310.png)
總在直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033355089487.png)
上. 12分
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,
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兩點.
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033109728389.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033109869373.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033109900500.png)
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橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240327315461149.png)
與雙曲線
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032731609645.png)
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(1)求橢圓E的方程;
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已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847354334.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847370636.png)
,雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847386372.png)
的左、右焦點分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847354334.png)
的左、右頂點,而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847386372.png)
的左、右頂點分別是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847354334.png)
的左、右焦點。
(1)求雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847386372.png)
的方程;
(2)若直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847464700.png)
與橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847354334.png)
及雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847386372.png)
都恒有兩個不同的交點,且L與的兩個焦點A和B滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847510655.png)
(其中O為原點),求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031847542313.png)
的取值范圍。
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科目:高中數(shù)學
來源:不詳
題型:單選題
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824034240264433.png)
,則方程
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824034240295826.png)
表示的曲線不可能是( )
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科目:高中數(shù)學
來源:不詳
題型:單選題
已知拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033838276544.png)
上一點P到y(tǒng)軸的距離為5,則點P到焦點的距離為( )
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科目:高中數(shù)學
來源:不詳
題型:填空題
橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033631354724.png)
內(nèi)有一點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033631370535.png)
,過點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033631385289.png)
的弦恰好以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824033631385289.png)
為中點,那么這條弦所在直線的斜率為
,直線方程為
.
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