1 . 已知椭圆方程为
,离心率为
且过点
.
(1)求椭圆方程;
(2)过左焦点
的直线交椭圆于
两点,是否存在实数
,使
恒成立?若存在,求此时
的最小值;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/d5c7316976a221c051a2c14df80b1347.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/310f780f4f03699023b1322a1e002539.png)
(1)求椭圆方程;
(2)过左焦点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f8a6fb8b04355296310391bdec8a934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e4ea7acf140086ae816a4025d2683dc.png)
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解题方法
2 . 在平面直角坐标系中,点
在运动过程中,总满足关系式
.
(1)求点
的轨迹
的方程;
(2)过点
作两条斜率分别为
的直线
和
,分别与
交于
和
,线段
和
的中点分别为
,若
,证明直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c62b58e1ce45cfd3fe723345eaf411f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17aa130296d594a23b0a7a864fc33320.png)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd3b260036958c271fee22820b05fdb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4f5fac15de56be6dfb7ba2429b54cea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d762c4e0c2e788c94066aeea1530f4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/227c1d105f7abf228e7a4f3097ae93f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2026c8a047f60c7b84f4078466dcce6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/077aaf808a6243d4af30a3eb9320fb99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42887d9bf31c1dd99f13c39e63c9ab9.png)
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4卷引用:河南省驻马店市新蔡县第一高级中学2023-2024学年高二下学期6月月考数学试题
解题方法
3 . 已知椭圆C
的左、右焦点分别为
,过点P(2,1)的直线l与C交于A,B两点,当直线l过
时,直线l的斜率为
,且
的周长为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/706e6c63b376e3a7cf45a2b475c4f0b4.png)
(1)求C的方程;
(2)若过点A且斜率为
的直线交C于另外一点D,证明:直线BD恒过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b68bed5f53ce98a19785d81b7ca2336.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47cfe4e08c06bde245e58aa22485044c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5eb2485f90dbfd0dfd6e7d179a856f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/706e6c63b376e3a7cf45a2b475c4f0b4.png)
(1)求C的方程;
(2)若过点A且斜率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3389f53711264b0acba3ba6019f8b908.png)
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4 . 已知椭圆
的左、右焦点分别为
,上顶点为
是椭圆
在第一象限上的点,满足
.
(1)求椭圆
的方程;
(2)过直线
上的一点
,作椭圆
的两条切线
,切点分别为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c8f9a340b852b3682d9d85b54fce40e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c7e61cd4c81de00d8509e3dd04b9620.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc5749165ec0dc435539deda60fb5496.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(2)过直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdbd8a5d973b7a54b7605388fdcfbb07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/228b02b73e5333584ebe1e84d21ec6a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e1e4115d78e625e9e0f47cdade3286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6b556e2aff1b00b66c177a002e5cc3e.png)
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解题方法
5 . 已知椭圆
过点
,直线
过
的上顶点和右焦点,
的倾斜角为
,且满足
.
(1)求椭圆
的标准方程;
(2)设
两点为椭圆
的左、右顶点,点
(异于左、右顶点)为椭圆
上一动点,直线
,
的斜率分别为
,
,求证:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/310f780f4f03699023b1322a1e002539.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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/340a629d168688f0b5c5a82292c974c6.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.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/c5db41a1f31d6baee7c69990811edb9f.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/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b881044b5c73db6fcce110525741b02.png)
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2024-04-29更新
|
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2卷引用:河南省2023-2024学年高二下学期期中联考数学试题
名校
解题方法
6 . 已知椭圆的焦点为
,且该椭圆经过点
.
(1)求椭圆的标准方程;
(2)直线
过
且与椭圆交于
两点,当
面积最大时,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fecb84eefb8a3ed8db2ce10acae6b6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22760845eeb2bdc4d546584670d175ac.png)
(1)求椭圆的标准方程;
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ff8a5886e42095da57422c8777c10d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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2024-04-19更新
|
370次组卷
|
2卷引用:河南省商丘市2023-2024学年高二下学期5月联考数学试题
名校
解题方法
7 . 已知椭圆
:
的左、右焦点分别为
,
,过点
作x轴的垂线与椭圆交于M,N两点,
,
.
(1)求椭圆C的标准方程;
(2)若椭圆C的上顶点为P,直线l与该椭圆交于A,B两点(异于上、下顶点),记直线PA的斜率为
,直线PB的斜率为
,且
,证明:直线l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/48da128547c4cf9745e8e4b99988a3db.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/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cbdd945cdadb7dca0d281d791374573.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3f1e02c7c258970e28c4854c36c9dc8.png)
(1)求椭圆C的标准方程;
(2)若椭圆C的上顶点为P,直线l与该椭圆交于A,B两点(异于上、下顶点),记直线PA的斜率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ec7bcf5820dfe70290259c2d7ac1ea5.png)
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2024-04-07更新
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2卷引用:河南省部分学校(金科)大联考2023~2024学年高二下学期第一次质量检测数学试题
名校
解题方法
8 . 已知椭圆
:
的短轴长为
,且椭圆
经过点
.
(1)求椭圆
的方程;
(2)若过点
的直线与椭圆
相交于
,
两点,且
,求直线
的方程.
![](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/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7950b79b4e43d650ca5d84e9fce65540.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23baf026a1dd9532845c195a428d8e76.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/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc7df99fe6438442a9453fc0c57fb703.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
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2024-04-07更新
|
654次组卷
|
3卷引用:河南省洛阳市2023-2024学年高二上学期期末考试数学试题
名校
解题方法
9 . 已知椭圆C:
(
,
)的长轴为
,短轴长为4.
(1)求椭圆C的标准方程;
(2)设直线l:
与椭圆C交于不同两点A、B,且
,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41322821ce31416fdac8dd6e0aa41c71.png)
(1)求椭圆C的标准方程;
(2)设直线l:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54479885d4ab2f717d2e97718da04b43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83f423f4c2973942ab64731bc81c40bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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2024-03-31更新
|
1673次组卷
|
2卷引用:河南省南阳市桐柏县2023-2024学年高二上学期期末质量检测数学试题
名校
解题方法
10 . 已知椭圆
的左、右焦点分别为
,过
上的一点
作
的切线
,点
关于
的对称点分别为
,则四边形
的面积为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77d98da0e2532648a4435045ae73de2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e67baac84cf5c95d06d50c36cab7c68.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/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13d726b14a313e000a81526162f35748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a8b843637b5e9c963b7520b4fe8340.png)
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