1 . 已知椭圆
和
的离心率相同,设
的右顶点为
,
的左顶点为
,
,
(1)证明:
;
(2)设直线
与
的另一个交点为P,直线
与
的另一个交点为Q,连
,求
的最大值.
参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dca5a95b0a6c43bb6fd7bd97678a29cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5e05710167d91511ad0b58921e4219a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49cba284d675a3028d7a8d54f1f8ae70.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2890cea2b175baf0e214726fa3233bf6.png)
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b7d857811cbd619f868d951aa7a0ab8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57831b59ea828a088ed9daec8babb774.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d44e8bc37ed03f44470762748a8f942a.png)
参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/019125de7ac580e95b0295b235b23179.png)
您最近一年使用:0次
2 . 已知椭圆
的离心率为
,且椭圆上一点与椭圆两个焦点构成的三角形的周长为
,
(1)求椭圆
的方程.
(2)设直线
与椭圆交于
两点,若以
为直径的圆经过椭圆的右顶点
,求
的值及
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c6162828793e697cb1ad643b287c4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24c14ff9b66f21c05e52dc3c8908c2df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3643fa70165ed9fef1398bc9443e7a02.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbf46f98449e6b3cd06d129d003ae32f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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3 . 已知椭圆
,椭圆
的焦点在y轴上.经过点
且与椭圆
有相同的离心率.
(1)求椭圆
的方程;
(2)设A为椭圆
的上顶点,点P是椭圆
上在第一象限内的一点,点Q与点P关于原点对称,直线
与椭圆
的另一个交点分别为M,N两点,设
与
的面积分别为
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be052f0deaa8c6de37c57ad17edd32a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecd80b108b9a6cf2ef20bab203674831.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(2)设A为椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5671fb25040a712a49e8c8148d67d300.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98013a5042685a1db94249e70c62c09a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9763846b1131e1e3e2d741ad95d5bb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3637753af5ce86be9c23a9beb6b5067.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235f0a6fb218d28383e6f27f2df1f50f.png)
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4 . 国家体育场“鸟巢”的钢结构如图所示,内外两圈的钢骨架是离心率相同的椭圆,若由外层椭圆长轴一端点A和短轴一端点B分别向内层椭圆引切线AC,BD(如图),且两切线斜率之积等于
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/2/449388e5-2f52-4781-aef9-612c04d34667.jpg?resizew=350)
(1)若内层椭圆方程为
,求外层的椭圆方程;
(2)求椭圆的离心率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af6ddf31b7d9225a4239883af72d153b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/2/449388e5-2f52-4781-aef9-612c04d34667.jpg?resizew=350)
(1)若内层椭圆方程为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
(2)求椭圆的离心率.
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解题方法
5 . 已知椭圆
:
(
)的长轴长4,离心率
,
(1)求椭圆
的方程;
(2)设
,
分别为椭圆
左,右顶点,已知点
为直线
:
上的动点,直线
、
与椭圆
分别交于
、
两点,求证:直线
经过定点,并求出该定点的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5de85df85401e7e8da683ea4a784963c.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
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解题方法
6 . 已知经过原点O的直线与离心率为
的椭圆
交于A,B两点,
、
是椭圆C的左、右焦点,且
面积的最大值为1.
![](https://img.xkw.com/dksih/QBM/2021/3/16/2679239932182528/2682916238000128/STEM/4a32f349-91e5-40db-a236-e22936e21d32.png)
(1)求椭圆C的标准方程;
(2)如图所示,设点P是椭圆C上异于左右顶点的任意一点,过点Р的椭圆C的切线与
交于点M.记直线
的斜率为
,直线
的斜率为
,证明:
为定值,并求出该定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2cfd997d3b66a3b8f7731b26f0ab0c8.png)
![](https://img.xkw.com/dksih/QBM/2021/3/16/2679239932182528/2682916238000128/STEM/4a32f349-91e5-40db-a236-e22936e21d32.png)
(1)求椭圆C的标准方程;
(2)如图所示,设点P是椭圆C上异于左右顶点的任意一点,过点Р的椭圆C的切线与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639c3d2ff5ee566fcc1b69c65712a661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ac86e1c253297a377e14fb9a1689be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/183b6a0cef4256c9696a5bca31053da5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b881044b5c73db6fcce110525741b02.png)
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解题方法
7 . 如图所示,椭圆
的离心率为
,过点
作直线
交椭圆于不同两点
,
.
![](https://img.xkw.com/dksih/QBM/2020/5/14/2462494403534848/2463252372111360/STEM/85300749-e9a2-4a0b-a186-bc111ff4e46b.png)
(1)求椭圆的方程;
(2)①设直线的斜率为
,求出与直线
平行且与椭圆相切的直线方程(用
表示);
②若
,
为椭圆上的动点,求四边形
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21b71f5213c5e23d621a80d545f0a289.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d56ab70e602f2e2e291df643ab209162.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://img.xkw.com/dksih/QBM/2020/5/14/2462494403534848/2463252372111360/STEM/85300749-e9a2-4a0b-a186-bc111ff4e46b.png)
(1)求椭圆的方程;
(2)①设直线的斜率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
您最近一年使用:0次
名校
8 . 已知:点
是离心率为
的椭圆
:
上的一点.斜率为
的直线BD交椭圆C于B、D两点,且A、B、D三点不重合.
(Ⅰ)求椭圆C的方程;
(Ⅱ)△ABD的面积是否存在最大值?若存在,求出这个最大值;若不存在,请说明理由?
(Ⅲ)求证:直线AB、AD的斜率之和为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d58aab742dcee01ea15f2a74fbee0c4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fb223b8777ab973970491bf0dcc6806.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
(Ⅰ)求椭圆C的方程;
(Ⅱ)△ABD的面积是否存在最大值?若存在,求出这个最大值;若不存在,请说明理由?
(Ⅲ)求证:直线AB、AD的斜率之和为定值.
您最近一年使用:0次
2017-07-01更新
|
900次组卷
|
3卷引用:江苏省苏州市第五中学2016-2017学年高二12月月考数学试题
9 . 已知椭圆C:
的离心率为
,且C上任意一点到两个焦点的距离之和都为4.
(Ⅰ)求椭圆C的方程;
![](https://img.xkw.com/dksih/QBM/2015/5/4/1572093589946368/1572093596106752/STEM/48f8803d5b3f47909c09ebc71eacd4dd.png)
(Ⅱ)设直线
与椭圆交于P、Q,O为坐标原点,若
,求证
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ec56b59d6f2654570c2b5c4fd13a05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
(Ⅰ)求椭圆C的方程;
![](https://img.xkw.com/dksih/QBM/2015/5/4/1572093589946368/1572093596106752/STEM/48f8803d5b3f47909c09ebc71eacd4dd.png)
(Ⅱ)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ec9ac3c9a223321dcba6e5df6d4328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/470b43996302afddc155f87e63a96775.png)
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10 . 已知椭圆C:
离心率
,短轴长为
.
(Ⅰ)求椭圆
的标准方程;
(Ⅱ) 如图,椭圆左顶点为A,过原点O的直线(与坐标轴不重合)与椭圆C交于P,Q两点,直线PA,QA分别与y轴交于M,N两点.试问以MN为直径的圆是否经过定点(与直线PQ的斜率无关)?请证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ec56b59d6f2654570c2b5c4fd13a05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/075ba8c6fb5ef7288cd3fed425c8e69e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95bacae35b6e16a0a33c2bdc6bc07df7.png)
(Ⅰ)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(Ⅱ) 如图,椭圆左顶点为A,过原点O的直线(与坐标轴不重合)与椭圆C交于P,Q两点,直线PA,QA分别与y轴交于M,N两点.试问以MN为直径的圆是否经过定点(与直线PQ的斜率无关)?请证明你的结论.
![](https://img.xkw.com/dksih/QBM/2015/5/4/1572091436417024/1572091442601984/STEM/a541436e799e4c118d2fe67fe92ff423.png)
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