1 . 参数方程是以参变量为中介来表示直线或曲线上点的坐标的方程,是直线或曲线在同一坐标系下的另一种表现形式.很多曲线(如心脏线、螺线、玫瑰线)都可以用参数方程呈现.在平面直角坐标系
中,直线
的参数方程式
(
为参数),其中
,角
为直线
的倾斜角.曲线
的参数方程是
(
为参数).其中
,直线
与曲线
相交于
、
点.
(1)根据以上的参数方程求出直线
的一般式方程和曲线
的标准方程;
(2)设点
,设点
对应的参数为
,试证明:
;
(3)试问是否存在角
,使得对于任意的点
,表达式
均为定值
,若存在,请求出
及值
(结果用
,
表示);若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9b1cf149172b6c4a6526b25aba683be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1d5e2dfa2d5b134c85995877eff156b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73dd51ce19cf9b0ebfa8e42190c72bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.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/e77eee60e92c3e08a5877062cd1e925f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a990942b9fa26d28cee8579325da3675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(1)根据以上的参数方程求出直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93bb2baf350ed7e3490fd9e7399ce5c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d9fd58e71dcae6cafaf9037d20ebd76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39b1c2f6f5103b4a981e417b620dd239.png)
(3)试问是否存在角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93bb2baf350ed7e3490fd9e7399ce5c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6df16c0ff148acd2c4eac082120e43be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/291f17141e5dfbb8e129a9e59d23c120.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
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2 . 在极坐标系中,曲线
的方程是:
,且与
、
轴正半轴交于
、
两点.点
为曲线
上任意一点,将
绕原点逆时针旋转
,且长度变为原来的一半,得到点
,点
的轨迹为曲线
.射线:
与曲线
交于点
,与曲线
交于点
.以极点为原点,极轴为
轴建立直角坐标系.
(1)求直线
的一个参数方程及曲线
的极坐标方程;
(2)求线段
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/371ed9a8f6647b919b73f4741513e900.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.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/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d49f8a63ddbca52039fa9ab44cda6b29.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/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17f5d2e8186f0173d7862b1d39fb3dc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)求线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c07a88d8187839a9ca38041a406a405.png)
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3 . 第十四届全国冬季运动会于2月17日在内蒙古呼伦贝尔开幕,这是继北京冬奥会后全国举办的又一冬季项目大型体育赛事,也是内蒙古首次承办的全国大型综合体育盛会.本次赛事共设8个大项,16个分项,176个小项.在开闭幕期间,运动员、裁判员、教练员、媒体记者等总规模达4000余人.武大靖、任子威等明星运动员也纷纷亮相.某高中体育爱好者打算借四叶草具有幸福幸运的象征意义,准备设计一枚四叶草徽章以作纪念.如图,在极坐标系
中,方程
表示的图形为“四叶草”对应的曲线
.
的
时;求以极点为圆心的单位圆与
的交点的极坐标;
(2)设
和
是
上的两点,且
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/374d388d837e6e7e845e1e45dd3943b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a9e1dd70ccd7718f7ede19005034cad.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/e38fec745f18e1c06ecd27a5f6b2577f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56150248dd4b787a2013311e4737e93f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8a99d2c4a23825f62aadcc40822b5eb.png)
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解题方法
4 . 对于椭圆
,令
,
,那么在坐标系
中,椭圆经伸缩变换得到了单位圆
,在这样的伸缩变换中,有些几何关系保持不变,例如点、直线、曲线的位置关系以及点分线段的比等等;而有些几何量则等比例变化,例如任何封闭图形在变换后的面积变为原先的
,由此我们可以借助圆的几何性质处理一些椭圆的问题.
(1)在原坐标系中斜率为k的直线l,经过
,
的伸缩变换后斜率变为
,求k与
满足的关系;
(2)设动点P在椭圆
上,过点P作椭圆
的切线,与椭圆
交于点Q,R,再过点Q,R分别作椭圆
的切线交于点S,求点S的轨迹方程;
(3)点
)在椭圆
上,求椭圆上点B,C的坐标,使得△ABC的面积取最大值,并求出该最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8429aec72d26401b12a55b8337261df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/070cb835e194f9bb99aba9daf58bd2b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50443405ab95a95149c68f59f96619de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/863b5f9f0a7c6b7956979a5abc76d8d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e08c4a230e32f550374a5fa4db5f204.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/848d4055ca831ecde46d1b666ba9e33d.png)
(1)在原坐标系中斜率为k的直线l,经过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/070cb835e194f9bb99aba9daf58bd2b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50443405ab95a95149c68f59f96619de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbc8ced3660dab6e343773fd9dccebc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbc8ced3660dab6e343773fd9dccebc3.png)
(2)设动点P在椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/841307fdcdbbccacd07b652db535631f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce30fc0664cca88dbe6d38f32aee81e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd76519af3c3a098a590ad302acc003b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aaafb050b24c4e806c480e0665aaa5a.png)
(3)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad492d5033448d419df9c9b75a71894e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/271e595c257e4c0ade90a9bbbf0e6b0d.png)
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5 . 极坐标方程
,
和
所表示的曲线围成的面积为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1656178bb8d085a5cbeda8e90cf4e86e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f148770f94443d3720c0ec10f998989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/841afc4d922ade4abfae0bceeaae425f.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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6 . 已知点
与
关于坐标原点对称,则
等于( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbff397ced62d95bc9c1563d5145d4d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d14e520be9f58ff30eff40b10ceb0d82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88584cf1df43e28d03592c7998b1653.png)
A.5 | B.1 | C.![]() | D.![]() |
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2023-10-23更新
|
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2卷引用:江苏省宿迁市泗阳县实验高级中学2023-2024学年高二上学期开学测试数学试题
7 . 瑞士数学家雅各布·伯努利在1694年类比椭圆的定义,发现了双纽线.双纽线的图形如图所示,它的形状像个横着的“8”,也像是无穷符号“∞”.定义在平面直角坐标系
中,把到定点
距离之积等于
的点的轨迹称为双纽线
.以
为极点,
轴的正半轴为极轴建立极坐标系.
(1)求双纽线
的极坐标方程;
(2)双纽线
与极轴交于点P,点M为C上一点,求
面积的最大值(用
表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1aa1a5565eef09e163e2b3487beaa6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5abd0f31afe865a63682ccd4a18a0e0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/26/5dede47c-56c9-4ba0-b918-1a0fc4f70c20.png?resizew=179)
(1)求双纽线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)双纽线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a51268fce97426487c3338d6ec3d571.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2023-05-20更新
|
629次组卷
|
4卷引用:江西省重点中学协作体2023届高三第二次联考数学(理)试题
名校
解题方法
8 . 数学中有许多美丽的曲线,例如曲线
,(t为参数)的形状如数字8(如图),动点A,B都在曲线E上,对应参数分别为
与
,设O为坐标原点,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/9/444a2eeb-2633-42e1-ab9d-6f3ed4fb2909.png?resizew=149)
(1)求C的轨迹的参数方程;
(2)求C到坐标原点的距离d的最大值和最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a11d6a96018681bcf04768dc8d7e6601.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1adb726efaa7daa1613f56a6d75da819.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0248ff66368b4c849bc98c7d86dc644.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bc170fc6c893de77fdbb1d5a9b34814.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/9/444a2eeb-2633-42e1-ab9d-6f3ed4fb2909.png?resizew=149)
(1)求C的轨迹的参数方程;
(2)求C到坐标原点的距离d的最大值和最小值.
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2023-05-08更新
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4卷引用:河南省豫南名校毕业班2023届高三仿真测试三模理科数学试题
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9 . 已知曲线C的极坐标方程为
,A,B是曲线C上不同的两点,且
,其中O为极点.
(1)求曲线C的直角坐标方程;
(2)求点B的极径.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/293bcfd0679b28bea188755ae9c10d9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f375e6d640ae99446982f05221810ce3.png)
(1)求曲线C的直角坐标方程;
(2)求点B的极径.
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2023-04-23更新
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443次组卷
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4卷引用:四川省成都市第七中学2022-2023学年高二下学期期中考试数学(理)试题
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10 . “太极图”是关于太极思想的图示,其形状如对称的阴阳两鱼互抱在一起,也被称为“阴阳鱼太极图”.在平面直角坐标系
中,“太极图”是一个圆心为坐标原点,半径为
的圆,其中黑、白区域分界线
,
为两个圆心在
轴上的半圆,
在太极图内,以坐标原点为极点,
轴非负半轴为极轴建立极坐标系.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/24/cf8fafc0-b3cf-4970-98ad-3f6807fab2f1.png?resizew=196)
(1)求点
的一个极坐标和分界线
的极坐标方程;
(2)过原点的直线
与分界线
,
分别交于
,
两点,求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.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/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c54373fa62652ad075f41280e44a81b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/24/cf8fafc0-b3cf-4970-98ad-3f6807fab2f1.png?resizew=196)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)过原点的直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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5卷引用:江西省南昌市2023届高三二模数学(理)试题
江西省南昌市2023届高三二模数学(理)试题江西省南昌市2023届高三二模数学(文)试题四川省南部中学2023届高三下学期高考考前理科数学模拟训练(一)(已下线)专题20坐标系与参数方程(已下线)专题20坐标系与参数方程.