橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240209545481085.png)
的左、右焦點分別為F
1(-1,0),F(xiàn)
2(1,0),過F
1作與x軸不重合的直線l交橢圓于A,B兩點.
(I)若ΔABF
2為正三角形,求橢圓的離心率;
(II)若橢圓的離心率滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954564706.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954579292.png)
為坐標原點,求證:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954595698.png)
.
(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954626385.png)
;(Ⅱ)見解析.
試題分析:(Ⅰ)由橢圓定義易得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954642399.png)
為邊
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954657374.png)
上的中線,在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954689631.png)
中,可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954704534.png)
,即得橢圓的離心率;(Ⅱ)設
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954720585.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954735616.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954751664.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954782264.png)
,先得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954798636.png)
,再分兩種情況討論,①是當直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954829500.png)
軸垂直時;②是當直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954845396.png)
不與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954860266.png)
軸垂直時,都證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954891607.png)
,可得結(jié)論.
試題解析:(Ⅰ)由橢圓的定義知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954907762.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954923614.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954938574.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954642399.png)
為邊
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954657374.png)
上的中線,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954985571.png)
, 2分
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954689631.png)
中,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955032852.png)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954704534.png)
,∴橢圓的離心率
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954626385.png)
. 4分
(注:若學生只寫橢圓的離心率
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954626385.png)
,沒有過程扣3分)
(Ⅱ)設
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954720585.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954735616.png)
因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954751664.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954782264.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954798636.png)
6分
①當直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954829500.png)
軸垂直時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955313661.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955328612.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240209553281118.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955344605.png)
=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955360818.png)
,因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955375732.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954891607.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955391486.png)
恒為鈍角,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955422195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955438730.png)
. 8分
②當直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954845396.png)
不與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954860266.png)
軸垂直時,設直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020954845396.png)
的方程為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955640601.png)
,代入
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955672769.png)
,
整理得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240209556871052.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955703893.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955718909.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955734827.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955765979.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955781883.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240209557961393.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240209558281021.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240209558431053.png)
10分
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955859730.png)
,由①可知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955874575.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955391486.png)
恒為鈍角.,所以恒有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020955921681.png)
. 12分
練習冊系列答案
相關(guān)習題
科目:高中數(shù)學
來源:不詳
題型:解答題
已知拋物線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024009440525.png)
的焦點為F
2,點F
1與F
2關(guān)于坐標原點對稱,以F
1,F
2為焦點的橢圓C過點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024009455645.png)
.
(Ⅰ)求橢圓C的標準方程;
(Ⅱ)設點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024009471304.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024009487479.png)
,過點F
2作直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024009518280.png)
與橢圓C交于A,B兩點,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024009533694.png)
,若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024009565997.png)
的取值范圍.
查看答案和解析>>
科目:高中數(shù)學
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)
的長軸長為4,且過點
![](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
![](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,若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952295959.png)
,點
![](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,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952357313.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021952373315.png)
兩點的坐標分別為
![](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)
.
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021048010999.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021048025473.png)
為其右焦點,離心率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021048041307.png)
.
(Ⅰ)求橢圓C的標準方程;
(Ⅱ)若點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021048056562.png)
,問是否存在直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021048072613.png)
,使
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021048088253.png)
與橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021048103284.png)
交于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021048119488.png)
兩點,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021048134964.png)
.若存在,求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021048150290.png)
的取值范圍;若不存在,請說明理由.
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
如圖,在平面直角坐標系
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651466449.png)
中,橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651498993.png)
的右焦點為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651513478.png)
,離心率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651513387.png)
.
分別過
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651529293.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651544301.png)
的兩條弦
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651560374.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651576392.png)
相交于點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651591311.png)
(異于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651607291.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651622299.png)
兩點),且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651638507.png)
.
(1)求橢圓的方程;
(2)求證:直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651654398.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020651669380.png)
的斜率之和為定值.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240206517005862.png)
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:單選題
已知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021424429440.png)
分別是橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240214244441129.png)
的左右焦點,過
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021424460334.png)
垂直與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021424476275.png)
軸的直線交橢圓于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021424491423.png)
兩點,若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021424507592.png)
是銳角三角形,則橢圓離心率的范圍是( )
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:填空題
已知雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020811900752.png)
的離心率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020811916552.png)
,頂點與橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020811916713.png)
的焦點相同,那么雙曲線的焦點坐標為_____;漸近線方程為_________.
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
已知橢圓C的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020315281997.png)
,其離心率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020315312319.png)
,經(jīng)過橢圓焦點且垂直于長軸的弦長為3.
(Ⅰ)求橢圓C的方程;
(Ⅱ)設直線l:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020315328890.png)
與橢圓C交于A、B兩點,P為橢圓上的點,O為坐標原點,且滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020315344623.png)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824020315359423.png)
的取值范圍.
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546889313.png)
的兩個焦點為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546904719.png)
,點
![](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
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013546967549.png)
,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)
上任一點,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013547029548.png)
的取值范圍.
查看答案和解析>>