1 . 如图所示,已知半圆O的直径为
,l为位于半圆之外,而又垂直于
延长线的一直线,其垂足为T,且
,又M,N是半圆上的不同的两点,
,
,且
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dea2ae9d515f9ab351ad72306b776ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e7acd656195631c58a22b060c0d3ab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f5903d670301ba4abbfda6324be1a30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68d9c9b2c01681fab7d312271a860e46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8bec9ebcbb0a854dfe16b90a7894c9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab14817a0fe0920d012bf0d81ede4f0a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/28/0658cae7-9fee-442b-a262-8750ef39329d.png?resizew=188)
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2021-09-25更新
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228次组卷
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2卷引用:高中数学解题兵法 第三十三讲 命题之间的转化与变换
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解题方法
2 . 在新中国成立70周年国庆阅兵庆典中,众多群众在脸上贴着一颗红心,以此表达对祖国的热爱之情.在数学中,有多种方程都可以表示心型曲线,其中有著名的笛卡尔心型曲线.如图,在直角坐标系中,以原点O为极点,x轴正半轴为极轴建立极坐标系.图中的曲线就是笛卡尔心型曲线,其极坐标方程为
,M为该曲线上的任意一点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/a032092a-3f11-4bbc-a62d-87bf35df53f7.png?resizew=157)
(1)当
时,求M点的极坐标:当M的极角为
时,求它的极径;
(2)若过极点的直线
与该曲线相交于两点A,B,求证:弦长
为定值,并求出这个定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad655f4f1fe74ae7f018fd0ef2711a36.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/a032092a-3f11-4bbc-a62d-87bf35df53f7.png?resizew=157)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a07205bc511042fdda781b84436fb157.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f99376e6b053ffd21b179108c18c3d3.png)
(2)若过极点的直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4dfec890cdfdda355e19463f3be813.png)
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2021-07-24更新
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840次组卷
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4卷引用:贵州省贵阳市第一中学2021届高三下学期高考适应性月考卷(五)数学(理)试题
贵州省贵阳市第一中学2021届高三下学期高考适应性月考卷(五)数学(理)试题贵州省贵阳市第一中学2021届高三下学期高考适应性月考卷(五)数学(文)试题(已下线)专题22 坐标系与参数方程-备战2022年高考数学(理)母题题源解密(全国甲卷)(已下线)专题22 坐标系与参数方程-备战2022年高考数学(文)母题题源解密(全国甲卷)
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3 . 在平面直角坐标系中,P为曲线
(
为参数)上的动点,将P点纵坐标不变,横坐标变为原来的一半得Q点.记Q点轨迹为
,以坐标原点
为极点,
轴非负半轴为极轴建立极坐标系.
(1)求证:曲线
的极坐标方程为
;
(2)
是曲线
上两点,且
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9d0427cf88f3b650c59cc6778ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)求证:曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/549ea649efbff82eb432cdcc5cdd73b9.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9faea4deadc69fec42afab055f464340.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43e651a7faef2db0444fe157613b5779.png)
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2021-03-06更新
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1092次组卷
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3卷引用:陕西省西安中学2021届高三下学期第二次模拟考试数学(文)试题
4 . 在平面直角坐标系xOy中,曲线C:
经过伸缩变换
后所得曲线记为
.以O为极点,x轴的正半轴为极轴,建立极坐标系Ox.
(Ⅰ)求曲线
的极坐标方程;
(Ⅱ)已知A,B是曲线
上任意两点,且
,求证:O到直线AB的距离为常数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5f5d967ad135991b6075ee45df55643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0d61dedf0459a7719067c8d4d148dad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
(Ⅰ)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
(Ⅱ)已知A,B是曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3825ccc273ef9a672a606432d165b866.png)
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5 . 在平面直角坐标系
中,直线
的参数方程为
(
为参数且
).在以坐标原点为极点,
轴的正半轴为极轴的极坐标系中,曲线
的极坐标方程为
.
(1)求直线
的极坐标方程及曲线
的直角坐标方程;
(2)若点
在直线
上,点
在曲线
上,求证:
.
![](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/fe9de280d224eb4972da6c7fa2de3fb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/823ab696d27d40920c39b8c910789380.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/897b95b4fa201f0e08bff76016cf2fd9.png)
(1)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若点
![](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/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a440a9d51215d37f046d1bb411b55aa.png)
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6 . 在平面直角坐标系xOy中,动点P(x,y)的坐标满足
(t为参数),以原点O为极点,x正半轴为极轴建立极坐标系,曲线l的极坐标方程为ρsin(θ+φ)=cosφ(其中φ为常数,且φ
)
(1)求动点P的轨迹C的极坐标方程;
(2)设直线l与轨迹C的交点为A,B,两点,求证:当φ变化时,∠AOB的大小恒为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4094ed48901dd2d813eb6ac07bf9bcc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/254822ff878fcd710db6523c8727490c.png)
(1)求动点P的轨迹C的极坐标方程;
(2)设直线l与轨迹C的交点为A,B,两点,求证:当φ变化时,∠AOB的大小恒为定值.
您最近一年使用:0次
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7 . 已知直线
的参数方程为
(
为参数,
),以坐标原点为极点,以
轴正半轴为极轴建立极坐标系,曲线
的极坐标方程为
,射线
,
分别与曲线
交于
三点(不包括极点
).
(Ⅰ)求证:
;
(Ⅱ)当
时,若
两点在直线
上,求
与
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea0c89112e34d5096a06165c6cbd015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10adc3992efb2badc23f81d36faa206f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/925b34a1e7a48f450d883f9a0197e5c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0be5c8e545a1251a276a7726d469f25a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db2388a1c07fdee1a7accfb6882e93c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38335830b93ac4d99c28a8e209eecb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(Ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59fec83a7cecffd20f3fbd85dba6bc4d.png)
(Ⅱ)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91950023a67715fe5b61f1c9d1d6f87f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da6b93dbe5272a5167ff4e2918bec864.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
您最近一年使用:0次
2018-03-14更新
|
1106次组卷
|
5卷引用:广东省六校(广州二中,深圳实验,珠海一中,中山纪念,东莞中学,惠州一中)2018届高三下学期第三次联考数学(理)试题