解题方法
1 . 对于一个三维空间,如果一个平面与一个球只有一个交点,则称这个平面是这个球的切平面.已知在空间直角坐标系
中,球
的半径为
,记平面
、平面
、平面
分别为
、
、
.
(1)若棱长为
的正方体、棱长为
的正四面体的内切球均为球
,求
的值;
(2)若球
在
处有一切平面为
,求
与
的交线方程,并写出它的一个法向量;
(3)如果在球面上任意一点作切平面
,记
与
、
、
的交线分别为
、
、
,求
到
、
、
距离乘积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5e336d6ca2cae3d6e6c3810d7e521a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddd48d1ef9e8cd3b7aea60fd95b70fb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ef1e8a88d934eca5399decc64fdbd43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f435efcc7869eec21bdba1ed81dc3f5.png)
(1)若棱长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2122e3f1e76a635e58e4d54aa594c552.png)
(2)若球
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f5dcb10e84c60bbb67a382349ebeb3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c67202feb9b75fb893e9fc70cc1059d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c67202feb9b75fb893e9fc70cc1059d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
(3)如果在球面上任意一点作切平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f435efcc7869eec21bdba1ed81dc3f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
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名校
2 . 中国古代数学名著《九章算术》中记载:“刍甍者,下有袤有广,而上有袤无广.刍,草也.甍,屋盖也.”翻译为“底面有长有宽为矩形,顶部只有长没有宽为一条棱.刍甍是茅草屋顶.”现有一个刍甍如图所示,四边形ABCD为正方形,四边形ABFE,CDEF为两个全等的等腰梯形,
,
,
,
.
(1)当点N为线段AD的中点时,求证:直线
平面EFN;
(2)当点N在线段AD上时(包含端点),求平面BFN和平面ADE的夹角的余弦值的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e41d3f7d55fcbaebc4e2450ac63a3dc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed6034301fc4110da89bdb0f46ad82ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b578af6297446dfbf9fd7924b75adaef.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/24/8f13343a-e8f9-4b74-931e-d9afd12785c4.png?resizew=213)
(1)当点N为线段AD的中点时,求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca5dd496ee0c1170ef6dcc48266ee444.png)
(2)当点N在线段AD上时(包含端点),求平面BFN和平面ADE的夹角的余弦值的取值范围.
您最近一年使用:0次
2023-09-24更新
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1451次组卷
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11卷引用:辽宁省部分名校2023-2024学年高二上学期联考数学试题
辽宁省部分名校2023-2024学年高二上学期联考数学试题四川省眉山市青神县青神中学校2023-2024学年高二上学期期中数学试题广东省广州市第四中学2023-2024学年高二上学期期中数学试题广东省广州市三校(南实、铁一、广外)2023-2024学年高二上学期期中联考数学试题陕西省西安中学2023-2024学年高二上学期11月期中数学试题(已下线)难关必刷01 空间向量的综合应用-【满分全攻略】2023-2024学年高二数学同步讲义全优学案(人教A版2019选择性必修第一册)(已下线)特训02 期末解答题汇编(第1-5章,精选38道)-2023-2024学年高二数学《重难点题型·高分突破》(人教A版2019选择性必修第二册)吉林省东北师范大学附属中学2023-2024学年高二上学期第一次月考数学试题广东五校2022-2023学年高二下学期期末联考数学试题广东省佛山市顺德德胜学校2023-2024学年高二上学期期中数学试题山东省济宁市兖州区2023-2024学年高二上学期期中考试数学试题
名校
解题方法
3 . 设常数
.在棱长为1的正方体
中,点
满足
,点
分别为棱
上的动点(均不与顶点重合),且满足
,记
.以
为原点,分别以
的方向为
轴的正方向,建立如图空间直角坐标系
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/13/e06e4e4f-445b-442f-8e80-1e0f640affc9.png?resizew=217)
(1)用
和
表示点
的坐标;
(2)设
,若
,求常数
的值;
(3)记
到平面
的距离为
.求证:若关于
的方程
在
上恰有两个不同的解,则这两个解中至少有一个大于
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eb6bf23a5a12e1ba5413594d7b1a57c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a91c73ae980263c97742283b6b5852a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea77ba313fcc751481ac1ca214df3fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e85b55b6ad43be1a03fc637e1d3429.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/651066b6919cab279373a8a1e1130839.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d68873c59a21b0cd408cdf2b47d51096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a14c388e1e2e5a2ff1ccf6caffbee0d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/13/e06e4e4f-445b-442f-8e80-1e0f640affc9.png?resizew=217)
(1)用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15b268af571f9ecb37a864a08862814.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/200f24e682c93e02a87f3f9d57dc5d40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d262b77e692c60e3c6b6afb610e8fe66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c28b88046022376b082b8a45c04577c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a90170d7ef5ff6d1d63517c166f7a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a19e72906b84a1cb049167afdebdce0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
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4 . 如图,四棱锥
的底面
为直角梯形,平面
平面
,
,
,
,
,
.
(1)若三棱锥
的外接球的球心恰为
中点,求
与平面
所成角的正弦值;
(2)求四棱锥
体积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf9a6db3571fa57bfa2d5e4d44c51b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f79863ffcfa63117ca6741b20a48e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17bc77b37986d658edad69992c5ea0c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19fc9f894312e55c87a0d6737080e233.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4994abbead410afe78b1d4e9858aaba.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/28/43afedc7-0414-4fc6-b556-4bdbd81e78b8.png?resizew=127)
(1)若三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
(2)求四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
您最近一年使用:0次
2023-07-27更新
|
994次组卷
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4卷引用:浙江省台州市名校联盟2022-2023学年高二上学期11月五科联赛数学试题
浙江省台州市名校联盟2022-2023学年高二上学期11月五科联赛数学试题(已下线)模块三 专题1 利用空间向量求解探究性问题和最值问题(已下线)专题01 空间向量与立体几何(5)重庆市2024届高三上学期9月月度质量检测数学试题
21-22高二·湖南·课后作业
5 . 已知
的三个顶点分别为
,
,
.求证:
是直角三角形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b67f00fc8cbfe77d14a99c5de7479ec4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64f037d42f43dc1ea25242fadde956ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ad66c8b940c22a7e255052b74b6de4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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名校
6 . 已知椭圆
:
(
,
)的左、右焦点分别为
、
,离心率为
,经过点
且倾斜角为
的直线
与椭圆交于
、
两点(其中点
在
轴上方),
的周长为8.
(1)求椭圆
的标准方程;
(2)如图,将平面
沿
轴折叠,使
轴正半轴和
轴所确定的半平面(平面
)与
轴负半轴和
轴所确定的半平面(平面
)互相垂直.
,求异面直线
和
所成角的余弦值;
②是否存在
,使得折叠后
的周长为
?若存在,求
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](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/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b202870415ad76362476c8f122729d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5eb2485f90dbfd0dfd6e7d179a856f5.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)如图,将平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0530e3288e75edc196427ebc1448f201.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16498e054295750f17b6fb4c05f66b84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bc1f08c7640e62e8717abf4d44a6c83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cabea664e61863b3b3279dbce607924e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3656055f5256cd06e636ea96e9f89c2.png)
②是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b202870415ad76362476c8f122729d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5eb2485f90dbfd0dfd6e7d179a856f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7163395f9aaa29be7f6b3106ba48b744.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43660b1543b3a2b46185f7629d28a963.png)
您最近一年使用:0次
2021-08-16更新
|
2975次组卷
|
9卷引用:浙江省杭州学军中学紫金港校区2021-2022学年高二上学期期中数学试题
浙江省杭州学军中学紫金港校区2021-2022学年高二上学期期中数学试题2023年全国中学生数学能力测评(终评)高三年级组试题河北省部分学校2024届高三上学期摸底考试数学试题(已下线)第三章 折叠、旋转与展开 专题一 平面图形的翻折、旋转 微点6 圆锥曲线中的翻折问题(一)2024届河北省部分高中高考一模数学试题(已下线)专题07 直线与圆、圆锥曲线湖南省长沙市明德中学2023-2024学年高二下学期3月阶段测试数学试卷福建省厦门市集美中学2020-2021学年高二下学期第一次月考数学试题甘肃省白银市靖远县第一中学2024届高三下学期模拟预测数学试题
7 . 已知椭圆
的左、右焦点分别为
、
.经过点
且倾斜角为
的直线
与椭圆
交于
、
两点(其中点
在
轴上方),
的周长为8.
(1)求椭圆
的标准方程;
(2)如图,把平面
沿
轴折起来,使
轴正半轴和
轴确定的半平面,与
负半轴和
轴所确定的半平面互相垂直.
,求异面直线
和
所成角的大小;
②若折叠后
的周长为
,求
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9bece414af7ecb2d796dc8a6f549e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27e62a44b8712ce4483b8710cda0dc1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d49bc14700562fee13af6a548e4eaef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5e56b5cbcf69756792fc934bcc20cb.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
(2)如图,把平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bc1f08c7640e62e8717abf4d44a6c83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cabea664e61863b3b3279dbce607924e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3656055f5256cd06e636ea96e9f89c2.png)
②若折叠后
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5e56b5cbcf69756792fc934bcc20cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7163395f9aaa29be7f6b3106ba48b744.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
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