名校
1 . 已知双曲线
的一条渐近线方程为
,焦距为6,左顶点为
,点
是双曲线
的右支上相异的两点,直线AB,AC分别与直线
交于点
,且以线段
为直径的圆恰过双曲线
的右焦点
.
(1)求双曲线
的标准方程;
(2)求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ef66f4832adc43902055a7e6d258037.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25869dd14f3e7412beda491bb83f982d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab609a6574633ebabcff3e73fa862081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40460e97733a56b0b9963f8c641c47c3.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/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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2 . 已知双曲线C:
的右顶点为M,过点
的直线l交双曲线C于A,B两点,设直线MA的斜率为
,直线MB的斜率为
.
(1)求直线l斜率的取值范围;
(2)证明:
为定值,并求出该定值;
(3)求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1892b7c3cd7bea116f532f66fba44662.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f13bf66fc845b115de4ec45b4be0e23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
(1)求直线l斜率的取值范围;
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b69e3f7ddd51215d00661c09cd900d60.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7384b9afcef2d86a87eee0c66f383052.png)
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解题方法
3 . 已知双曲线
的右焦点为
,点
在双曲线
上,
.
(1)若
,且点
在第一象限,点
关于
轴的对称点为
,求直线
与双曲线
相交所得的弦长;
(2)探究:
的外心是否落在双曲线
在点
处的切线上,若是,请给出证明过程;若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2b30352c43707c4e54b94ce5b61f2e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/021bcc5ea186cd32c39b3d333b0f448c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0888ec49f9bba4ae0ec0ff57423ca50e.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/761d73623dcfb06f436844101786d71e.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/902f97913e1af1e6c793f7edfe6b2114.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)探究:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5adccd1dd14171c8c29d4a3836728c0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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2024-03-21更新
|
768次组卷
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3卷引用:吉林省白山市2024届高三第二次模拟考试数学试题
4 . 已知双曲线C的中心为坐标原点,左焦点为
,离心率为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/28/c40da5f2-1ed7-48ed-a8a9-6307c332b0ec.png?resizew=197)
(1)求C的方程;
(2)记C的右顶点为A,过点A作直线MA,NA与C的左支交于M,N两点,且
,
,D为垂足.证明:存在定点Q,使得
为定值,并求出Q点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e47e3ecdff18bdbe66c366f19aeac1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/28/c40da5f2-1ed7-48ed-a8a9-6307c332b0ec.png?resizew=197)
(1)求C的方程;
(2)记C的右顶点为A,过点A作直线MA,NA与C的左支交于M,N两点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3342ceb71dc21a5e6150caad2ce0b029.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7920d2550a6af7df3db60a33fe02c53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad6c1e83db94aee2d609d5736d985eac.png)
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5 . 已知双曲线
的离心率为
,且其焦点到渐近线的距离为1.
(1)求
的方程;
(2)若动直线
与
恰有1个公共点,且与
的两条渐近线分别交于
两点,
为坐标原点,证明:
的面积为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28dbdb64f3a81099b5ce93f4871c8bd3.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若动直线
![](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/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea1f0417d8269f01d8e0bc1a8756e2ac.png)
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2024-01-17更新
|
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6卷引用:吉林省通化市梅河口市第五中学2023-2024学年高二上学期期末数学试题
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解题方法
6 . 已知直线
与双曲线
无公共交点,则C的离心率的取值范围是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3710b5f6d5c02114ab1cb3cd2acc3c4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
您最近一年使用:0次
2023-11-09更新
|
1468次组卷
|
11卷引用:吉林省通化市梅河口市第五中学2023-2024学年高二下学期开学考试数学试题
吉林省通化市梅河口市第五中学2023-2024学年高二下学期开学考试数学试题江苏省苏州市苏州实验中学2023一2024学年高二上学期12月质量检测数学试题四川省成都市树德中学2023-2024学年高二下学期入学考试数学试卷河北省邯郸市永年区第二中学等校2023-2024学年高二上学期期中联考数学试题(已下线)重庆市巴南区重庆实验中学校2023-2024学年高二上学期期中数学试题江苏省苏州市常熟中学2023-2024学年高二上学期12月阶段性学业水平调研数学试题四川省成都市天府新区实外高级中学2023-2024学年高二上学期百人计划第二次段考数学试题四川省绵阳市南山中学实验学校2023-2024学年高二上学期期末模拟数学试题(三)河南省信阳市商城县上石桥高级中学2023-2024学年高二上学期12月月考数学试题四川省成都市武侯高级中学2023-2024学年高二上学期12月月考数学试题安徽省阜阳市临泉第一中学(高铁分校)2023-2024学年高二上学期第三次月考数学试题
名校
解题方法
7 . 已知双曲线C:
的左、右焦点分别为
,
,且
,若C上的点M满足
恒成立.
(1)求C的方程;
(2)若过点M的直线l与C的两条渐近线交于P,Q两点,且
.
(i)证明:l与C有且仅有一个交点;
(ii)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3040b6c904477030ecf8ba20b2b18759.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/1d4f7e7f33963df24d6a46067b4677e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cdb28eafa72ace7027ac32d7dae0906.png)
(1)求C的方程;
(2)若过点M的直线l与C的两条渐近线交于P,Q两点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac480bb785925885de84d8284c428830.png)
(i)证明:l与C有且仅有一个交点;
(ii)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b206caf5ac005c8df99c13b258d3e179.png)
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2023-08-05更新
|
564次组卷
|
4卷引用:吉林省通化市梅河口市第五中学2023-2024学年高二下学期第一次月考数学试题
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8 . 已知双曲线
的左顶点为
,过
的直线
与
的右支交于点
,若线段
的中点在圆
上,且
,则双曲线
的离心率为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/41b96eda0601673fafb836643969914f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e41477cb9bc876e2c5d01b1f971f9af0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
A.![]() | B.![]() | C.2 | D.3 |
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2023-07-27更新
|
1398次组卷
|
3卷引用:吉林省实验中学2023-2024学年高二上学期期末考试数学试卷
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解题方法
9 . 已知双曲线![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a5bbb709522dba9425a6b45ee671298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f3fa0b40fb0d9b8c62e37316ab3b04.png)
经过
,
两点.
(1)求C的标准方程;
(2)若直线
与C交于M,N两点,且C上存在点P﹐满足
,求实数t的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a5bbb709522dba9425a6b45ee671298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f3fa0b40fb0d9b8c62e37316ab3b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f311053d11884b1a21d5f9b5724996c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/448c0a5ee776d19ce8e42ac9a5fd27c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/177ae4fa30cba0272d338973b8f7bdf6.png)
(1)求C的标准方程;
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82c87531c3e3346663fe16c8eff34956.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/167b2d17e1f2b2ceac8edb8918636280.png)
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2023-02-13更新
|
610次组卷
|
3卷引用:吉林省吉林市吉林毓文中学2023-2024学年高二上学期期末考试数学试题
吉林省吉林市吉林毓文中学2023-2024学年高二上学期期末考试数学试题(已下线)高二数学开学摸底考01(江苏专用)-2023-2024学年高中下学期开学摸底考试卷山东省济南市2022-2023学年高二下学期期末数学试题