解题方法
1 . 如图所示,在棱长为1的正方体
中,点
为截面
上的动点,若
,则点
的轨迹长度是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2f7554a52815bfa0f4d75221ba7397.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39ccd5ca2447dc8825f0e54e0ad873aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
A.![]() | B.![]() | C.![]() | D.1 |
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4卷引用:贵州省毕节市赫章县乌蒙山学校教育集团2023-2024学年高一下学期5月联考数学试题
贵州省毕节市赫章县乌蒙山学校教育集团2023-2024学年高一下学期5月联考数学试题山东省潍坊市2024届高三一模数学试题2024届山东省滨州市一模联考数学试题(已下线)专题1 立体几何中的截面问题【练】(1)
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解题方法
2 . 已知椭圆C:
(
)的离心率为
,左顶点A到右焦点
的距离为3.
(1)求椭圆
的方程;
(2)设直线
与椭圆
交于不同两点
,
(不同于A),且直线
和
的斜率之积与椭圆的离心率互为相反数,求
在
上的射影
的轨迹方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设直线
![](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/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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
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贵州省贵阳市第一中学2023-2024学年高三上学期高考适应性月考(四)(12月)数学试题江西省赣州市南康中学2024届高三上学期七省联考考前数学猜题卷(九)(已下线)专题18 圆锥曲线高频压轴解答题(16大核心考点)(讲义)-1
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解题方法
3 . 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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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
4 . 在正四棱台
中,
,点
在四边形
内,且
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b893c7da2b26d7576a05bb6a887b2887.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9df53c3a9b1c3691d8f216fd3139763.png)
A.正四棱台![]() |
B.正四棱台![]() ![]() |
C.正四棱台![]() ![]() |
D.![]() ![]() |
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解题方法
5 . 阿波罗尼斯是古希腊著名数学家,与欧几里得、阿基米德被称为亚历山大时期数学三巨匠,阿波罗尼斯发现:平面内到两个定点
的距离之比为定值
,且
的点的轨迹是圆,此圆被称为“阿波罗尼斯圆”.在平面直角坐标系
中,
,点
满足
.设点
的轨迹为曲线
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/844da174d67557f7a44c0962d51189b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e56a9870ee77ce3930c6692af93bc7dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddf1db3d608d81245f34a0d7b1aaab2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df5d30e4268a4b86a4e098e8cb57da3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
A.![]() ![]() |
B.点![]() ![]() |
C.当![]() ![]() |
D.若![]() ![]() ![]() |
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6 . 已知圆心在
轴上移动的圆经过点
,且与
轴、
轴分别交于
、
两个动点,记点
.
(1)求
的轨迹方程;
(2)若直线
与曲线
交于
、
两点,
为坐标原点,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a3738bb3fced399273145e04ec3955b.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/858edb9a2fe9bab726479ec89a0f72c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a71b0db6cc0131e0e40f2ba38019458.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d259822ab64b8626f3893b8432673358.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/139189550dc26d6130088799c1e65581.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.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/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea1f0417d8269f01d8e0bc1a8756e2ac.png)
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7 . 已知
为坐标原点,
,过动点
作直线
的垂线,垂足为点
,
.记动点
的轨迹曲线为
.已知
,
,
,
均在
上,直线
,
的唯一交点为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16415383f0112bf97da824e970c1d299.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bb7ca2d544f528ff1afcbb5c365f337.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c23f5f351c545d42078775e430805e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/042f6277c98e108cab95992342e4bfd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e26184bcf7695a20d4d85444d852a04c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e8b2f3657868c619dc476bcf77adf5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b57d2dd0af3a6061a2280fef3373950d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc34876d748f30fa4fc2eb6a686b5ff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3b119d0cf10a948cdc53c1066af0b8b.png)
A.曲线![]() ![]() |
B.![]() |
C.![]() |
D.若![]() ![]() ![]() ![]() ![]() ![]() |
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解题方法
8 . 如图,在平面直角坐标系
中,直线
与
轴交于点
,过
右侧的点
作
,垂足为
,且
.
(1)求点
的轨迹
的方程;
(2)过点
的动直线
交轨迹
于
,设
,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b556b1a9944719cf423e90f8df16c773.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bdd3be8e58a9150e5b1e2a43988fd82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee166bece50bbf1fc8a647b69472d39a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/6/bf9a8fb2-cd2f-48b8-8ca0-c7a66cb532c4.png?resizew=110)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f55d12701014cf53071093e8739d089b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fe7fd9b0c3c203a053a7ea52b71e7c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b62ca4c0140614d0fa57362fa260e6b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e8293f682eeec40e39b4858d611d6f7.png)
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5卷引用:贵州省贵阳市观山湖区第一高级中学2022-2023学年高二下学期第二次月考数学试题
贵州省贵阳市观山湖区第一高级中学2022-2023学年高二下学期第二次月考数学试题广东省清远市阳山县南阳中学2023-2024学年高二上学期第二次月考(期中)数学试题四川省成都市第七中学2023届高考热身文科数学试题(已下线)模块一 情境6 以解析几何为背景(已下线)第06讲 3.3.2抛物线的简单几何性质(2)
9 . 已知动点
到点
和点
的距离之比为
,若至少存在3个点
到直线
:
的距离为
,则
的取值范围为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19b62194097ac66a5093c57fca2f5b4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e95e84f5c91c910aaafc5e74dbfbdf59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5881f1ce9b4172ca346032d0fd1e3d.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/a6fb5f07d7214180cef2d4bc24aa785a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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10 . 已知O为坐标原点,M是椭圆
上的一个动点,点N满足
,设点N的轨迹为曲线
.
(1)求曲线
的方程.
(2)若点A,B,C,D在椭圆
上,且
与
交于点P,点P在
上.证明:
的面积为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f036026cd92e9ad059c3f22a7658638.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22234a15c785f14348e5e912308b5b28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(2)若点A,B,C,D在椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5f7a87f605c77a237facc0d05425887.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/177678001b2ccde1db8f57fa5e017002.png)
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2023-01-12更新
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10卷引用:贵州省贵阳清镇北大培文学校2022-2023学年高二下学期3月月考数学试题