名校
解题方法
1 . 已知双曲线
的两条渐近线分别为
和
,右焦点坐标为
为坐标原点.
(2)直线
与双曲线的右支交于点
(
在
的上方),过点
分别作
的平行线,交于点
,过点
且斜率为4的直线与双曲线交于点
(
在
的上方),再过点
分别作
的平行线,交于点
,这样一直操作下去,可以得到一列点
.
证明:①
共线;
②
为定值
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4073d527d4b14759a7cbaeabfb35a756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d9fc68d6d88485e65bfc599dfdde547.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c388c5a37852ee8ea1066ce1f9c1d5dd.png)
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81337e2e2dbdd0c3148454fa41d44144.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8454989732716850cb57ca15f8ef596.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8454989732716850cb57ca15f8ef596.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28c7d2c85e7878b6cbfb45b71ffb60b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68ee712dfc82e1acc31ef8dcad479a39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a71fc9c0068109dad1382354570665.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68ee712dfc82e1acc31ef8dcad479a39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28c7d2c85e7878b6cbfb45b71ffb60b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7ca7a25d804f483ec024c735c633e62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8357bec2e11a9f9756cd7c7e9143afca.png)
证明:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1db4fe79b98486f0d4556f37d64d74f9.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aeda3264e04e2d3ed2a15c69adad05c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47dd9cb5cac04bc15c02f292b38d2928.png)
您最近一年使用:0次
2024-05-08更新
|
481次组卷
|
3卷引用:辽宁省鞍山市第一中学2024届高三下学期八模数学试卷
解题方法
2 . 已知双曲线
:的右焦点为
,渐近线方程为
.
(1)求双曲线
的标准方程;
(2)设
为双曲线
的右顶点,直线
与双曲线
交于不同于
的
,
两点,若以
为直径的圆经过点
,且
于点
,证明:存在定点
,使
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8639f8fa866c17565dd4f75970665765.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d63f162c4846a76cadee56ae2f42e37c.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce50eeb654ef50f36a582c785f273ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/803a617fb53e67edbc2955cb629c329b.png)
您最近一年使用:0次
解题方法
3 . 已知双曲线
的一条渐近线方程是
,右焦点坐标为
.
(1)求
的方程;
(2)若过点
的直线
与
交于
,
两点(
,
均在
轴上方);线段
的中点为
,点
在线段
上,且满足
,设直线
,
(
为坐标原点)的斜率分别为
,
,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a2cfa22139b3e9c9a73500e1ba19f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08b9f0b9e53a83e68f5fec944f343119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/973aa6f1a268dbd02fa5236dc6f58647.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/328aaba77106396d4ca644c8b7a352e0.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.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/f52a58fbaf4fea03567e88a9f0f6e37e.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b4d6fdb6c4f177eda754f5e8fa6da0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e9f7d1272b7344346b58b660aa260a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.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)
您最近一年使用:0次
解题方法
4 . 双曲线
的一条渐近线方程为
,且经过点
.
(1)求
的方程;
(2)
为坐标原点,过双曲线
上一动点
(
在第一象限)分别作
的两条渐近线的平行线为
,
且
,
与
轴分别交于P,Q,求证:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45cc81cfaccc00aa4b7139de5a35a102.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35c06b4516b17a521106e8282300c31b.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bab77b1212086d7b16e288f73a09560.png)
您最近一年使用:0次
5 . 如图,已知双曲线
,
的左、右顶点恰是椭圆
的左、右焦点
、
,
的渐近线方程为
,
的离心率为
,分别过椭圆
的左右焦点
、
的弦
、
所在直线交于双曲线
上的一点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/28/2cfd3535-e8bf-4660-8b66-ffea66b07fb4.png?resizew=190)
(1)求
、
的标准方程;
(2)求证:
为定值;
(3)求证:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc7808754a3eb9be9f532d0393084936.png)
![](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/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d3051f43ac48c0a730a791b8a93ad37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.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/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/28/2cfd3535-e8bf-4660-8b66-ffea66b07fb4.png?resizew=190)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cfad0b9d319209620b5a5214fedd3cc.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1047d5e7dd5a6ec93119a867a191ff55.png)
您最近一年使用:0次
解题方法
6 . 已知双曲线
的焦距为8,双曲线
的左焦点到渐近线的距离为2.
(1)求双曲线
的方程;
(2)设
分别是双曲线
的左、右顶点,
为双曲线
上任意一点(
不与
重合),线段
的垂直平分线交直线
于点
,交直线
于点
,设点
的横坐标分别为
,求证:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)求双曲线
![](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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/782c9e617a2bd1391c2d7a29aec0bb73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f48d5ab4cc1d9ec97efa3e78d954be50.png)
您最近一年使用:0次
7 . 已知等轴双曲线C的中心为坐标原点O,焦点在x轴上,且焦点到渐近线的距离为
.
(1)求C的方程;
(2)若C上有两点P,Q满足
,证明:
是定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
(1)求C的方程;
(2)若C上有两点P,Q满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54edadd93c68b0105da2dccc8c3f410c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b547b0d2d050ef703d8bacfd49b32186.png)
您最近一年使用:0次
2023-03-26更新
|
819次组卷
|
6卷引用:辽宁省辽阳市2023届高考一模数学试题
8 . 在①C的渐近线方程为
②C的离心率为
这两个条件中任选一个,填在题中的横线上,并解答.
已知双曲线C的对称中心在坐标原点,对称轴为坐标轴,点
在C上,且______.
(1)求C的标准方程;
(2)已知C的右焦点为F,直线PF与C交于另一点Q,不与直线PF重合且过F的动直线l与C交于M,N两点,直线PM和QN交于点A,证明:A在定直线上.
注:如果选择两个条件分别解答,则按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d3051f43ac48c0a730a791b8a93ad37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
已知双曲线C的对称中心在坐标原点,对称轴为坐标轴,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13428e6305dfc0ca9883044f525b6b5a.png)
(1)求C的标准方程;
(2)已知C的右焦点为F,直线PF与C交于另一点Q,不与直线PF重合且过F的动直线l与C交于M,N两点,直线PM和QN交于点A,证明:A在定直线上.
注:如果选择两个条件分别解答,则按第一个解答计分.
您最近一年使用:0次
2023-01-14更新
|
769次组卷
|
5卷引用:辽宁省辽阳市协作校2022-2023学年高二上学期期末考试数学试题
辽宁省辽阳市协作校2022-2023学年高二上学期期末考试数学试题甘肃省庆阳市2022-2023学年高二上学期期末考试数学试题(已下线)2023年新课标全国Ⅱ卷数学真题变式题19-22(已下线)第04讲 3.2.2双曲线的简单几何性质(2)山东省烟台市龙口第一中学等校2023-2024学年高二上学期12月月考数学试题
名校
解题方法
9 . 已知双曲线
过点
,且C的渐近线方程为
.
(1)求C的方程.
(2)A,B为C的实轴端点,Q为C上异于A,B的任意一点,
与y轴分别交于M,N两点,证明:以
为直径的圆过两个定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41900c447140586d8f03c47a66127291.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6e0d4e3e322585fefe4c25020eda176.png)
(1)求C的方程.
(2)A,B为C的实轴端点,Q为C上异于A,B的任意一点,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92ca520748c8b8d3878fb112a89ada7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
您最近一年使用:0次
2022-01-16更新
|
540次组卷
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3卷引用:辽宁省辽阳市2021-2022学年高二上学期期末考试数学试题
2021高二·全国·专题练习
解题方法
10 . 已知双曲线C的渐近线方程为
,右焦点
到渐近线的距离为
.
(1)求双曲线C的方程;
(2)过F作斜率为k的直线
交双曲线于A、B两点,线段AB的中垂线交x轴于D,求证:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b6bb019e2d7c6d17d15ec4d9043f5e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b6fe0f38ab3095ac6575faa02914b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
(1)求双曲线C的方程;
(2)过F作斜率为k的直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13b8fbe288f426c55b7a72ed32d43272.png)
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