1 . 已知动点P到点
的距离等于其到直线
距离的2倍,记点P的轨迹为曲线
.
(1)求曲线
的方程;
(2)已知斜率为k的直线l与曲线
交于点
为坐标原点,若
,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63309dbc3612815f6dbdee23d9a10adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b650820d7bed48ed67a2869ad8c65ff1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)已知斜率为k的直线l与曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef644115c956ed62c3da8310c6f67ecd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3825ccc273ef9a672a606432d165b866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acb8956ebcde7cf0f1011a2fc1888e15.png)
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2023-12-25更新
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892次组卷
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4卷引用:山西省吕梁市孝义市2023-2024学年高二上学期12月月考数学试题
2022高三·全国·专题练习
2 . 已知双曲线
的右焦点为
是双曲线右支上一点,定点
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c58fa4a337f0b81b991fb32e8e6e3c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f83470bcabef39dae732060751132bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45e2b451fcf1c33c8cd4ecbfb1e15756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6166bfc653566f1797feb748aa735b30.png)
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3 . 已知双曲线
的左、右两个焦点分别为
是它左支上一点,
到左准线的距离为
,双曲线的一条渐近线为
,问是否存在点
,使
成等比数列?若存在,求出
的坐标;若不存在说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8c38260b151e844f75eb13190689c6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d9f00474d2a7ca8d683752474a0f8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45cc81cfaccc00aa4b7139de5a35a102.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3168f844701e7904b6a99c63eea39e58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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4 . 设
,
是双曲线
-
= 1
的左、右两个焦点,
为左准线,离心率
,
是左支上一点,P到
的距离为
,且
,| PF
|,| PF
|成等差数列,求此双曲线方程.
![](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/7355be4fcbc3130a5951364a3be76d6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5268413295580cfda0755ab458b36b64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d26d3f9d8344e1c727fbbed5421daaa2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d5d2b95ceced05d7bfab425ca3a2bc4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2484ed697198409a3c2bf52e68fbd255.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3edec6b3a624042e4295d9e709401b0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d68c9c1649a2fedf11b5b6358a898e6.png)
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5 . 一系列椭圆,经过
轴为公共准线,求椭圆右焦点的轨迹方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b87fa9e87c12a5e5b18e48a42c66a30c.png)
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解题方法
6 . 已知定点
,定直线
,动点
在曲线
上.
(1)设曲线
的离心率为
,点
到直线
的距离为
,求证:
;
(2)设过定点
的动直线与曲线
相交于
两点,过点
与直线
垂直的直线与
相交于点
,直线
是否过定点?若是,求出定点坐标;若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90ce47fde921058026708a4321a0e213.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af3a8183f5256dbe27e2fd7e3f4873e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22962a2ad892cb6b14ab039a06e8cdc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb4402aeb853b22f20992156957ef0fd.png)
(1)设曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0521d68f2fdb4ea414d75a85c3434f81.png)
(2)设过定点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.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/dad2a36927223bd70f426ba06aea4b45.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/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d7b816eca15d4b7d060013df53edd53.png)
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2024-03-01更新
|
433次组卷
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3卷引用:四川省大数据精准教学联盟2024届高三第一次统一监测理科数学试题
名校
解题方法
7 . 请阅读下列材料,并解决问题:
到一个定点
的距离和
到定直线
的距离的比是常数
,则动点的轨迹就是圆锥曲线(这个圆锥曲线的第二定义).其中定点
称为其焦点,定直线
称为其准线(其中椭圆与双曲线的准线方程为
,抛物线准线方程为
),正常数
称为其离心率.当
时,轨迹为椭圆;当
时,轨迹为抛物线;当
时,轨迹为双曲线.
(1)已知平面内的动点
到一个定点
的距离和
到定直线
的距离的比是常数
,则动点
的轨迹方程为 (直接写出结果,无需过程).
(2)在(1)所求的曲线中是否存在一点,使得该点到直线
的距离最小?最小距离是多少?
圆锥曲线的第二定义
二次曲线,即圆锥曲线,是由一平面截二次锥面得到的曲线,包括椭圆,抛物线,双曲线等.2000多年前,古希腊数学家最先开始研究二次曲线,并获得了大量的成果.古希腊数学家阿波罗尼斯采用平面切割圆锥的方法来研究二次曲线.阿波罗尼斯曾把椭圆叫“亏曲线”把双曲线叫做“超曲线”,把抛物线叫做“齐曲线”,事实上,二次曲线由很多统一的定义、统一的二级结论等等.比如:平面内的动点![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45ff7e0ef1f622120cc1b18e9d3e80ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f430f01710597c751d0766d7bc857596.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40ba597082f60b7382ccd7c8f4e6f7d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42b7ac29311c13aa538f3f48cb513b0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09dbcaa127022fbd6b6f13345196408a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a58c44592477e5cab15cd165ff9b3d78.png)
(1)已知平面内的动点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45ff7e0ef1f622120cc1b18e9d3e80ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8db3b46f0bf8897318fb3d0114e56e55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4300a7b81c82e20fe8bca7a453f8ff99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7294f5ae2a24ff42e84cd9773b2a7287.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)在(1)所求的曲线中是否存在一点,使得该点到直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faa97a6ae27b83f941b5c7e8350e7896.png)
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2023-12-28更新
|
487次组卷
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4卷引用:贵州省清镇市博雅实验学校2023-2024学年高二上学期第四次月考数学试题数学
贵州省清镇市博雅实验学校2023-2024学年高二上学期第四次月考数学试题数学重庆市万州二中教育集团2023-2024学年高二下学期入学质量监测数学试题(已下线)专题2 点点距离 构造函数 练(已下线)情境15 二级结论命题
8 . 已知点
到定点
和定直线
的距离之比是常数
,求点P的轨迹方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aee82283f06cedef32eb15b87964f5d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24550b13dbecf7d86c7054250e987274.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fa5d6092f598c7da4796f965e40525a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f374448976552eb59b33c0901e9bf938.png)
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23-24高二上·上海·课后作业
9 . 点
到定点
的距离与它到直线
的距离之比为
,求点
的轨迹方程,并说明此轨迹是什么图形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be92f0e0012a7696c78e3e00513edefd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/224d30ca84f1aeeeda7a718e751a4925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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真题
解题方法
10 . 已知一列椭圆.若椭圆
上有一点
,使
到右准线
的距离
是
与
的等差中项,其中
分别是
的左、右焦点.
(1)试证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b2b654f58cc444e3a86515692e46671.png)
(2)取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b950f01934279f704fae817976d19b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff06044e6d8c51fd92126e9634bcc255.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c360fec104715a3dde04e579cd08ee31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b36b9322eb3410a3dd5a7c99c054949.png)
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