名校
1 . 如图,已知椭圆
的左右顶点分别是
,
,焦点
,其中
,设点
,连接
交椭圆于点
,坐标原点是
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/11/6dfe213d-87f8-47a2-a26a-1a9da1464126.png?resizew=162)
(1)求椭圆的离心率;
(2)证明:
;
(3)设三角形
的面积为
,四边形
的面积为
,若
的最小值为1,求椭圆的标准方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](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/24550b13dbecf7d86c7054250e987274.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b05d3b8f5c9df891ef6fbcaf12f43207.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd81b31853a7d466db55dc2c6671f570.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/11/6dfe213d-87f8-47a2-a26a-1a9da1464126.png?resizew=162)
(1)求椭圆的离心率;
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58ac6979adfa3b30c6067a9fdfd49f08.png)
(3)设三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42f7c674f87a67fde99afceb3cfd8189.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ad5a9147b25285124851a61c7d1a24a.png)
您最近一年使用:0次
解题方法
2 . 已知椭圆
,一组平行直线的斜率是
.
(1)求这组直线何时与椭圆有两个公共点?
(2)当它们与椭圆有两个公共点时,证明这些直线被椭圆截得的线段的中点在同一条直线上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f64b7ab668d61bcb8ec6d195c672a81e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4b8503f4706b8321e4e79a87eadea84.png)
(1)求这组直线何时与椭圆有两个公共点?
(2)当它们与椭圆有两个公共点时,证明这些直线被椭圆截得的线段的中点在同一条直线上.
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解题方法
3 . 已知椭圆C:
过点
,且焦距为
.
(1)求C的方程;
(2)已知点
,
,E为线段
上一点,且直线
交C于G,H两点.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c179fe7eff7abfdd092b63c9c1b82d0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38387ba1cadfd3dfc4dea4ca9f613cea.png)
(1)求C的方程;
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e396c83557ec8b2b4b12d97d6738819.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b671cdde6baf9ab577330696ca8ff121.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a6c6e7c025362c46a64a8956761f08e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63038bd8a5e711b735d0af5a1ee6e655.png)
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解题方法
4 . 已知椭圆
的上顶点为
,右焦点为
,直线
与圆
相切.
(1)求椭圆
的方程;
(2)若不过点
的动直线
与椭圆相交于
,
两点,若
,求证:直线
过定点,并求出该定点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/069ffc1e3936d254303d588e1a70a3aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e374ab134118c55bc01540ebe5b0c1a1.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若不过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3630a0c2e68952b855a2246fcb96ddef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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名校
解题方法
5 . 在平面直角坐标系
中,已知椭圆
的离心率为
,且过点![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e90064d012356de1877aa697cd6d6ac.png)
(1)求椭圆C的方程;
(2)设直线l的斜率存在,不经过A点且与C交于两个不同的点P,Q,若直线
分别与y轴交于点
,且
,证明:直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e90064d012356de1877aa697cd6d6ac.png)
(1)求椭圆C的方程;
(2)设直线l的斜率存在,不经过A点且与C交于两个不同的点P,Q,若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5671fb25040a712a49e8c8148d67d300.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81778f5c4ab75f3f35ad0ee472fffe9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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6 . 给定椭圆
,称圆心在原点
,半径为
的圆是椭圆
的“伴随圆”.若椭圆
的一个焦点为
,其短轴上的一个端点到
的距离为
.
(1)求椭圆
的方程及其“伴随圆”方程;
(2)若倾斜角为
的直线
与椭圆
只有一个公共点,且与椭圆
的“伴随圆”相交于
两点,求弦
的长;
(3)在椭圆
的“伴随圆”上任取一点
,过点
作两条直线
,使得
与椭圆
都只有一个公共点,且
分别与椭圆的“伴随圆”交于
两点.证明:直线
过原点
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da001dad7941e6c9858637d7b62cec59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9eba828e0cec2057bc205fc42d64d370.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若倾斜角为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1e5fa72f2878b476bc57f0df12d6555.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86e203b7c9a6600e0272c58a23733490.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
(3)在椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f56635924584077092ac3c7dd68837b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
您最近一年使用:0次
2023-12-08更新
|
443次组卷
|
3卷引用:上海市复旦大学附属中学2023-2024学年高二上学期阶段性学业水平检测2(暨拓展考试6)数学试题
上海市复旦大学附属中学2023-2024学年高二上学期阶段性学业水平检测2(暨拓展考试6)数学试题四川省南充市南充高级中学2023-2024学年高二上学期第二次月考数学试题(已下线)第2章 圆锥曲线(压轴题专练)-2023-2024学年高二数学单元速记·巧练(沪教版2020选择性必修第一册)
7 . 已知椭圆
:
的右焦点为F(1,0),短轴长为2.直线
过点F且不平行于坐标轴,
与
有两个交点A,B,线段
的中点为M.
(1)求椭圆
的方程;
(2)证明:直线
的斜率与
的斜率的乘积为定值;
(3)延长线段
与椭圆
交于点P,若四边形
为平行四边形,求此时直线
的斜率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(3)延长线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14e902eb263971b466d0fcd91c56b453.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
您最近一年使用:0次
2023-12-08更新
|
1282次组卷
|
4卷引用:北京市第一六一中学2023-2024学年高二上学期12月阶段练习数学试题
解题方法
8 . 设椭圆
的离心率为
,上、下顶点分别为
.过点
,且斜率为
的直线
与
轴相交于点
,与椭圆相交于
两点.
(1)若
,求
的值;
(2)是否存在实数
,使得直线
平行于直线
?证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/827ccf0c04aa941ba20d5f4c6068b46b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9721fb5107049e7cd76adaac2fbb068.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/887e587a4fb083a37f3d84f42874ec16.png)
![](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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6114947179bed8c2c86ac078e2f8497f.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d12ea169e2633bd49074728867dea28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
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名校
解题方法
9 . 已知椭圆
的右焦点为
,点
为椭圆上一动点,且
到
的距离与到直线
的距离之比总是
.
(1)求椭圆
的方程;
(2)过
作椭圆
的切线,交直线
于点
.
①求证:
;
②求三角形
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdbd8a5d973b7a54b7605388fdcfbb07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67091bd26f940830395f4fe095b31031.png)
②求三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27a422884d9f6944de0b286439a114ec.png)
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2023-12-03更新
|
676次组卷
|
2卷引用:天津市南开中学2023-2024学年高二上学期第二次学情调查数学试卷
名校
解题方法
10 . 已知椭圆C:
的左、右焦点分别为
,点
是椭圆上不同于左右顶点的一动点,点
关于x轴的对称点为点
.当直线
过左焦点
时,
.
(1)求椭圆
的方程;
(2)若直线
与椭圆交于另外一点
(点
和点
不重合),证明直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5439f5ff9bd5deec0f0ef35c6f605b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/368fc197b61e01fe6a4a168bb7b375cd.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3656055f5256cd06e636ea96e9f89c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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