13-14高二下·上海金山·期末
1 . 下图是利用计算机作图软件在直角坐标平面
上绘制的一列抛物线和一列直线,在焦点为
的抛物线列
中,
是首项和公比都为
的等比数列,过
作斜率2的直线
与
相交于
和
(
在
轴的上方,
在
轴的下方).
证明:
的斜率是定值;
求
、
、
、
、
所在直线的方程;
记
的面积为
,证明:数列
是等比数列,并求所有这些三角形的面积的和.
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/b1fdd5907c8945daa1d2a6c5f85cff33.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/e5bc5a88b190402ea0a5585796983893.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/480e1c5e5b41471fb3cbb691b8dd760c.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/254ded389dcb47029ef772c5050ecdb8.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/9d29b9adb3cd4d17863ebcba6f543736.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/e5bc5a88b190402ea0a5585796983893.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/900a3ffa563046c689438590dba753a8.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/a564e5d460ff45f084c9f06f50d168aa.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/25671a533b2f49da9491f03489380d64.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/422941d4ca564a3a9450487377c153fa.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/25671a533b2f49da9491f03489380d64.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/5c35b050df974373881b650ad5208d3f.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/422941d4ca564a3a9450487377c153fa.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/5c35b050df974373881b650ad5208d3f.png)
证明:
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/ee9191fbd491406bb810d9de498be548.png)
求
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/d5fd4eafdb57476fa2e30c7b9df32b1c.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/194610503a754b7686aa315734c59db2.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/ff5ec7535416488e8cc40f3416673c07.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/25671a533b2f49da9491f03489380d64.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/ff5ec7535416488e8cc40f3416673c07.png)
记
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/4899235549084d7b8c01254c4f1bc148.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/e506fc2c8bb4417f818d405cf4ed084b.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/c8fdf56ddd71474da71fa61f9cd63e63.png)
![](https://img.xkw.com/dksih/QBM/2014/10/8/1571870225850368/1571870231740416/STEM/f9a43993371a48cfa376a82c6752a9ed.png)
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解题方法
2 .
为空间中两条互相垂直的直线,等腰直角三角形
的直角边
所在直线与
都垂直,斜边
以直线
为旋转轴旋转,有下列结论:
①当直线
与
成
角时,
与
成
角;
②当直线
与
成
角时,
与
成
角;
③直线
与
所成角的最小值为
;
④直线
与
所成角的最大值为
.
其中正确的是__________ (填写所有正确结论的编号)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
①当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
②当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
③直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a97bb4dcfab4ec7539bc783d563c49.png)
④直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
其中正确的是
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5卷引用:上海市七宝中学2021-2022学年高一下学期期末数学试题
上海市七宝中学2021-2022学年高一下学期期末数学试题北京市中国人民大学附属中学朝阳学校2021-2022学年高二10月月考数学试题(已下线)专题23空间点、线、面的位置关系-2022年高三毕业班数学常考点归纳与变式演练(文理通用)(已下线)专题25 盘点立体几何中最值问题——备战2022年高考数学二轮复习常考点专题突破(已下线)专题15 立体几何多选、填空题(理科)
名校
3 . a,b为空间两条互相垂直的直线,等腰直角三角形
的直角边
所在直线与a,b都垂直,斜边
以
为旋转轴选择,有下列结论:
①当直线
与a成60°角时,
与b成30°角;
②当直线
与a成60°角时,
与b成60°角;
③直线
与a所成角的最小值为45°;
④直线
与a所成角的最大值为60°;
其中正确的是_______.(填写所以正确结论的编号).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
①当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
②当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
③直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
④直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
其中正确的是_______.(填写所以正确结论的编号).
A.①③ | B.①④ | C.②③ | D.②④ |
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2020-01-10更新
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2卷引用:上海市上海中学2017-2018学年高二下学期期末数学试题
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4 . 设非零向量
是直线
的一个方向向量,则
可以是________ .(只需填写一个)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/337ab4dc61186ca58bb0605cb6d02df0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f19038b72902c839171a97e816583f21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/337ab4dc61186ca58bb0605cb6d02df0.png)
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5 . 定义:如果曲线段
可以一笔画出,那么称曲线段
为单轨道曲线 ,比如圆、椭圆都是单轨道曲线;如果曲线段
由两条单轨道曲线构成,那么称曲线段
为双轨道曲线 .对于曲线
有如下命题:
存在常数
,使得曲线
为单轨道曲线;
存在常数
,使得曲线
为双轨道曲线.下列判断正确的是( ).
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14884a97cc2e3321b7fb6f6de2f31fe2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51441c8788ff11be766766227793246d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce20ef9c08e82df8c7f45bac6dd31d36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
A.![]() ![]() | B.![]() ![]() |
C.![]() ![]() | D.![]() ![]() |
您最近一年使用:0次
2023-12-13更新
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9卷引用:上海市吴淞中学2023-2024学年高二上学期期末质量检测数学试卷
上海市吴淞中学2023-2024学年高二上学期期末质量检测数学试卷上海市同济大学第一附属中学2023-2024学年高二上学期期末考试数学试卷(已下线)专题03 圆 曲线与方程(九大题型+优选提升题)-【好题汇编】备战2023-2024学年高二数学下学期期末真题分类汇编(沪教版2020选择性必修,上海专用)上海市青浦区2024届高三上学期期终学业质量调研数学试题(已下线)第1章 坐标平面上的直线 (压轴题专练)-2023-2024学年高二数学单元速记·巧练(沪教版2020选择性必修第一册)(已下线)2.5 曲线与方程(五大题型)(分层练习)-2023-2024学年高二数学同步精品课堂(沪教版2020选择性必修第一册)(已下线)第2章 圆锥曲线(压轴题专练)-2023-2024学年高二数学单元速记·巧练(沪教版2020选择性必修第一册)(已下线)专题07 解析几何(三大类型题综合)15区新题速递(已下线)专题01 集合(15区真题速递)
6 . 已知双曲线C:
定义:把双曲线
的虚轴保持不变,渐近线的斜率变为原来渐近线斜率的两倍得到的曲线称为曲线
的“
线”,把双曲线
的左支向右平移
个单位,把它的右支向左平移
个单位得到的曲线称为曲线
的“
-线”,若双曲线
是等轴双曲线,且焦距等于
,
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/7/4c91a0eb-1442-46fa-bc3a-7a4cce048598.png?resizew=190)
(1)求双曲线
的“
-线”和“
-线”;
(2)若由“
-线”和“
-线”围成的封闭曲线上的点集都在圆
内或圆
上,求半径最小时圆
的方程,并在坐标系中用尺规作图画出该封闭曲线和圆
大致图像.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5ec7fa23be9cbe9a50607ea6bc8a4ff.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/4446f654cf8401b640cc79c2b97af5db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ec978eb43bc4f9e7df83b0d0195dcda.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/7/4c91a0eb-1442-46fa-bc3a-7a4cce048598.png?resizew=190)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
(2)若由“
![](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/ac047e91852b91af639feec23a9598b2.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/ac047e91852b91af639feec23a9598b2.png)
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7 . 我边防局接到情报,在海礁
所在直线
的一侧点
处有走私团伙在进行交易活动,边防局迅速派出快艇前去搜捕:如图,已知快艇出发位置在
的另一侧码头
处,
公里,
公里,
;
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/01ae3af3-0a60-478c-a57b-834262fb6aca.png?resizew=136)
(1)是否存在点
,使快艇沿航线
或
的路程相等;如存在,则建立适当的直角坐标系,求出点
的轨迹方程,且画出轨迹的大致图形;如不存在,请说明理由;
(2)问走私船在怎样的区域上时,路线
比路线
的路程短,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3c3f13203c1915b104924f650fe4227.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a54bf739ada87edb51304d38f09b46d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b15b82151bff7cc0238d2034a6129f0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/01ae3af3-0a60-478c-a57b-834262fb6aca.png?resizew=136)
(1)是否存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c24a1f5dc4d939fc7f6f76cfa387f73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a2391f20b822ea7fdde7b37a1713699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)问走私船在怎样的区域上时,路线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c24a1f5dc4d939fc7f6f76cfa387f73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a2391f20b822ea7fdde7b37a1713699.png)
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11-12高三·上海奉贤·期末
8 . 出租车几何学是由十九世纪的赫尔曼-闵可夫斯基所创立的.在出租车几何学中,点还是形如
的有序实数对,直线还是满足
的所有
组成的图形,角度大小的定义也和原来一样.直角坐标系内任意两点
定义它们之间的一种“距离”:
,请解决以下问题: (1)求点
、
的“距离”
;
(2)求线段
上一点
的距离到原点
的“距离”;
(3)定义:“圆”是所有到定点“距离”为定值的点组成的图形,点
求经过这三个点确定的一个“圆”的方程,并画出大致图象;(说明所给图形小正方形的单位是1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42dfb56b5bcb3c3954d7d65cc12fea49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae35beffd38dfb117b57804a9042b886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42dfb56b5bcb3c3954d7d65cc12fea49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cba69c334a66df7f45e382c1528c94ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba1f5820426a563fede050649122d945.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ca979687ffb2214e747525635a6912c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41b80ac9ca466eb94c7e0354f9cacb60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4dfec890cdfdda355e19463f3be813.png)
(2)求线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80cb4d2d461646bf1d49ec66f3304ec7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686be4dc7a016d6f67f9043960e46b3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93c7781652e183fefe7b0b8578da2735.png)
(3)定义:“圆”是所有到定点“距离”为定值的点组成的图形,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/187ddb68acf4efce9542d72cf5bd491f.png)
![](https://img.xkw.com/dksih/QBM/2012/1/28/1570701930340352/1570701935984640/STEM/e564d3079d0e4d96b17a82d5a842615a.png?resizew=334)
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