解题方法
1 . 已知椭圆C:
的焦距为2,
,
分别为其左,右焦点,过
的直线l与椭圆C交于M,N两点,
的周长为8.
(1)求椭圆C的方程;
(2)已知结论:若点
为椭圆C上一点,则椭圆C在该点的切线方程为
.点T为直线
上的动点,过点T作椭圆C的两条不同切线,切点分别为A,B,直线AB交x轴于点Q.证明:Q为定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.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/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4b5c81ee16e93e9822c4dc54c362cb3.png)
(1)求椭圆C的方程;
(2)已知结论:若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23b4f86e48e2b0d63c1865c60ed1e4d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a9656735f55e5de465e5667ba578d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/224d30ca84f1aeeeda7a718e751a4925.png)
您最近一年使用:0次
解题方法
2 . 已知椭圆
过点
,焦距为
.
(1)求椭圆
的方程;
(2)直线
:
与椭圆
交于异于
的两点
,直线
分别与直线
交于点
两点,
为坐标原点且
,求证:直线
过定点,并求出定点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bd056ad7b4674fe46f04643fe175538.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b10e8abf8690e4b129466ddb918bcc94.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15b256345d7109e081b7c895591e995d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f44755c5fee4b90266eac73ad47a128.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/564cc2c470001e7cd6fa28731a3875d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a089c207e39a24d0d82aa853ac2bbb8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5ad58997b9dc0b341c9af08f0cd1fbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/030c0325fde242e06cee8d270ba89d68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
您最近一年使用:0次
解题方法
3 . 已知椭圆
的左、右顶点分别为A、B,且
,点
在椭圆C上.
(1)求椭圆C的标准方程;
(2)若E,F为椭圆C上异于A,B的两个不同动点,且直线
与
的斜率满足
,证明:直线
恒过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57dfc9d1109fe41145cc892b5702d9fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ef233ad3db01fa3ce9ee94eaad8e64e.png)
(1)求椭圆C的标准方程;
(2)若E,F为椭圆C上异于A,B的两个不同动点,且直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df1161245861e8c6f72f713041cc3c41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
您最近一年使用:0次
2024-03-07更新
|
757次组卷
|
2卷引用:河南省部分重点高中2024届高三普通高等学校招生全国统一考试(期末联考)数学试卷
4 . 已知椭圆
过点
,焦距为
.过
作直线l与椭圆交于C、D两点,直线
分别与直线
交于E、F.
(1)求椭圆的标准方程;
(2)记直线
的斜率分别为
,证明
是定值;
(3)是否存在实数
,使
恒成立.若存在,请求出
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39ea1f5bdd213c7c3a571b4c38850bf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38387ba1cadfd3dfc4dea4ca9f613cea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76d03fa28c117649b0fdfe17eed7b583.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afeda89601db5178bf6f48eb93ef5820.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55aa0a20848c37c1892c567b2315e04.png)
(1)求椭圆的标准方程;
(2)记直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afeda89601db5178bf6f48eb93ef5820.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0002e26d1bc3291c9c0004ef0cf9c537.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7007702bb816d842023ff17e7e8a9d0e.png)
(3)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ba14f397c168b2e7fe26b5e38f1bb50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
2024-03-06更新
|
482次组卷
|
2卷引用:浙江省杭州学军中学紫金港校区2023-2024学年高二上学期期末数学试题
名校
解题方法
5 . 如图,已知椭圆
的短轴长为
,焦点与双曲线
的焦点重合.点
,斜率为
的直线
与椭圆
交于
两点.
的取值范围,并求椭圆
的方程.
(2)(本题可以使用解析几何的方法,也可以利用下面材料所给的结论进行解答)
极点与极线是法国数学家吉拉德·迪沙格于1639年在射影几何学的奠基之作《圆锥曲线论稿》中正式阐述的.对于椭圆
,极点
(不是原点)对应的极线为
,且若极点
在
轴上,则过点
作椭圆的割线交
于点
,则对于
上任意一点
,均有
(当斜率均存在时).已知点
是直线
上的一点,且点
的横坐标为2.连接
交
轴于点
.连接
分别交椭圆
于
两点.
①设直线
、
分别交
轴于点
、点
,证明:点
为
、
的中点;
②证明直线:
恒过定点,并求出定点的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0e1d8081ff8060cba32e46d280558a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a120cd30d1dd8cfac85539939c5febc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)(本题可以使用解析几何的方法,也可以利用下面材料所给的结论进行解答)
极点与极线是法国数学家吉拉德·迪沙格于1639年在射影几何学的奠基之作《圆锥曲线论稿》中正式阐述的.对于椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ee90e546232d08bb57108f2d5f87439.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbec8b46231e412ddce55cc96634e182.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0a4f3d3dbf83db17e1e0a2e16a15372.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8454989732716850cb57ca15f8ef596.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcfd3fc1c5776e964146498d94904045.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc854f8703f16bdbd8a30cd9f5d36bd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/790ef3382b1c731f2885eecfd92c2a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
①设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
②证明直线:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
您最近一年使用:0次
解题方法
6 . 已知椭圆
的离心率为
,左右顶点分别为A,B,G为C的上顶点,且
的面积为2.
(1)求椭圆C的方程;
(2)过点
的动直线与C交于M,N两点.证明:直线
与
的交点在一条定直线上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bc727c642cbc2181476b7dd8eca471e.png)
(1)求椭圆C的方程;
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00ed24bfcc37b79fe9ca61ed8fdf26ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.png)
您最近一年使用:0次
解题方法
7 . 已知椭圆
的离心率为
,依次连接四个顶点得到的图形的面积为
.
(1)求椭圆C的方程;
(2)过直线
上一点P作椭圆C的两条切线,切点分别为M,N,求证:直线
过定点.
![](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/41322821ce31416fdac8dd6e0aa41c71.png)
(1)求椭圆C的方程;
(2)过直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
您最近一年使用:0次
解题方法
8 . 椭圆
的离心率为
分别为椭圆的右顶点和上顶点,
为坐标原点,已知
面积为3.
(1)求椭圆
的标准方程;
(2)过点
作直线
交椭圆
于P,Q两点,过点
向
轴引垂线交MN于点B,点C为点P关于点B的对称点,求证:C,Q,M三点共线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e5578ca83f5bd5c285994061b9c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5894e8358f5d868a0e8547f8133b967.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/066ea7c8dac31105aadedad5f34d93fa.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ceec39aaaccbeaac9a4a5bdbb0f0e9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
您最近一年使用:0次
解题方法
9 . 已知圆
和定点
,
是圆
上任意一点,线段
的垂直平分线和半径
相交于点
,当点
在圆
上运动时,点
的轨迹记为曲线
.
(1)求曲线
的方程;
(2)直线
与曲线
交于不同两点
,
,直线
,
分别交
轴于
,
两点.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3cf9847c5060a6fcb659c0adfb3cb79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c6ff81aedbefa935da289dc632e78eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e4c6b0869d7248b6c2f964718a67702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48ad8115387158c8dda205d26968ba11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de33d0ffb95c8bd5971976e66629e53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.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/d064ea8cf59fea0a5d0faf0b6e4904ea.png)
您最近一年使用:0次
名校
解题方法
10 . 已知椭圆
:
的离心率为
,椭圆的一个顶点与两个焦点构成的三角形的面积为
.
(1)求椭圆
的方程;
(2)已知直线
与椭圆
相交于
、
两点,且与
轴,
轴交于
、
两点.
(i)若
,求
的值;
(ii)若点
的坐标为
,求证:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6115cadcab3c97b6413ab8642828598e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(i)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c5df68b7d1fd358239f30ab81fd6014.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(ii)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f519d25dbe037f78950dbda6170fef3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d871b0b46194d7300950e04f085533d2.png)
您最近一年使用:0次
2024-02-20更新
|
250次组卷
|
2卷引用:陕西省西安市第三中学2023-2024学年高二上学期期末数学试卷