1 . 在极坐标系中,
,
,
,以极点O为原点,极轴为x轴的正半轴,建立平面直角坐标系,已知直线1的参数方程为
( t为参数,
),且点P的直角坐标为
.
(1)求经过O,A,B三点的圆C的直角坐标方程;
(2)求证:直线l与(1)中的圆C有两个交点M,N,并证明
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19b62194097ac66a5093c57fca2f5b4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f93c06de0f8db44588f6e03bfb88bf3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70ef087f36061306cc6ffd37065850e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b682c1cf1c4eac10fdd3533b9f07a978.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb66f4db41478c23128adc14f2796556.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2c7b74fd862d7e3f35e40ae1f626c4c.png)
(1)求经过O,A,B三点的圆C的直角坐标方程;
(2)求证:直线l与(1)中的圆C有两个交点M,N,并证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79188647c574441c2414c3781a0ef543.png)
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6卷引用:贵州省贵阳市2021届高三上学期期末检测考试数学(理)试题
贵州省贵阳市2021届高三上学期期末检测考试数学(理)试题贵州省贵阳市普通中学2021届高三上学期期末监测考试数学(文)试题(已下线)专题29 坐标系与参数方程(解答题)-2021年高考数学(理)二轮复习热点题型精选精练(已下线)专题27 坐标系与参数方程(解答题)-2021年高考数学(文)二轮复习热点题型精选精练(已下线)专题15 坐标系与参数方程-备战2021届高考数学(文)二轮复习题型专练?(通用版)四川省绵阳南山中学2023届高三下学期4月绵阳三诊热身考试文科数学试题
解题方法
2 . 在直角坐标系
中,曲线C的参数方程为
(
为参数),以坐标原点为极点,x轴正半轴为极轴建立极坐标系,已知直线l的极坐标方程为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
.
(1)写出C的普通方程;
(2)写出直线l的直角坐标方程并判断l与C有无交点,如果有,则求出交点的直角坐标;如果没有,写出证明过程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e01f56efc66780e4bf66dc39f12fb27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b04652e01c3aecccc2d5053af869a657.png)
(1)写出C的普通方程;
(2)写出直线l的直角坐标方程并判断l与C有无交点,如果有,则求出交点的直角坐标;如果没有,写出证明过程.
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4卷引用:新疆叶城县第六中学2023届高三下学期高考考前最后一次诊断数学试题
3 . 在平面直角坐标系xOy中,曲线C的参数方程为
(
为参数).
(1)在以O为极点,x轴的正半轴为极轴的极坐标系中,求曲线C极坐标方程;
(2)若点A,B为曲线C上的两个点且
,求证:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac61ab188b14203f7cb4a1df244d0fda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6581916f5a65edfea257c804efee007e.png)
(1)在以O为极点,x轴的正半轴为极轴的极坐标系中,求曲线C极坐标方程;
(2)若点A,B为曲线C上的两个点且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3825ccc273ef9a672a606432d165b866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acb8956ebcde7cf0f1011a2fc1888e15.png)
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3卷引用:河南省南阳市2022-2023学年高三上学期期终质量评估(期末)数学(理)试题
河南省南阳市2022-2023学年高三上学期期终质量评估(期末)数学(理)试题河南省南阳市2022-2023学年高三上学期期终质量评估(期末)数学(文)试题(已下线)河南省南阳市2022-2023学年高三上学期期末数学(理)试题变式题21-23
4 . 在平面直角坐标
中,曲线
的参数方程为
(
为参数,
),以原点
为极点,
轴的正半轴为极轴建立极坐标系.
(1)求曲线
的普通方程和极坐标方程;
(2)在平面直角坐标
中,若过点![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
且倾斜角为
的直线
与曲线
交于
两点,求证:
成等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f346aec528524ea684d4652210a1e2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c51159984b2cb00f30b3986315019623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)在平面直角坐标
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/079dd115a4b8cbc93918a853363786dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037fb348109dc2063a268b10eb925a57.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/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/196591c97d9fe8a31f026253ea217050.png)
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4卷引用:四川省宜宾市2023届高三上学期第一次诊断性数学(理)数学试题
5 . 在平面直角坐标
中,以原点
为极点,
轴的正半轴为极轴建立极坐标系,曲线
的极坐标方程为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f1b2610e6d28fcd8a142e54946d3822.png)
(1)求曲线
的普通方程;
(2)在平面直角坐标
中,若过点![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
且倾斜角为
的直线
与曲线
交于
两点,求证:
成等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.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/9f1b2610e6d28fcd8a142e54946d3822.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)在平面直角坐标
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/079dd115a4b8cbc93918a853363786dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037fb348109dc2063a268b10eb925a57.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/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c1496b78d48796bc1f4a98bcb91df65.png)
您最近一年使用:0次
6 . 在平面直角坐标
中,曲线
的参数方程为
(
为参数,
),以原点
为极点,
轴的正半轴为极轴建立极坐标系.
(1)求曲线
的普通方程;
(2)在平面直角坐标
中,若过点![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
且倾斜角为
的直线
与曲线
交于
两点,求证:
成等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f346aec528524ea684d4652210a1e2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c51159984b2cb00f30b3986315019623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)在平面直角坐标
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/079dd115a4b8cbc93918a853363786dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037fb348109dc2063a268b10eb925a57.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/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/196591c97d9fe8a31f026253ea217050.png)
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7 . 在平面直角坐标系xOy中,曲线
的参数方程为
(
为参数),以坐标原点
为极点,
轴的正半轴为极轴,取相同长度单位建立极坐标系,直线
的极坐标方程为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbe6138f42d1fa54fd1d6fc643fa10d5.png)
(1)求曲线
的普通方程和直线
的直角坐标方程;
(2)设直线
与
轴的交点为
,经过点
的动直线
与曲线
交于不同的A,B两点,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b66a26751c662ca3bdf065b9e21cc73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbe6138f42d1fa54fd1d6fc643fa10d5.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e712d2bacb40498636e9ff78cb1df54.png)
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2卷引用:内蒙古自治区赤峰市赤峰二中2022-2023学年高三上学期第二次月考理科月考数学试题
名校
8 . 在平面直角坐标系
中,对于任意一点
,总存在一个点
满足关系式
,则称
为平面直角坐标系中的伸缩变换.
(1)在同一直角坐标系中,求平面直角坐标系中的伸缩变换
,使得椭圆
变换为一个单位圆;
(2)在同一直角坐标系中,
(
为坐标原点)经平面直角坐标系中的伸缩变换
得到
,记
和
的面积分别为
与
,求证:
;
(3)若
的三个顶点都在椭圆
上,且椭圆中心恰好是
的重心,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aee82283f06cedef32eb15b87964f5d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/495984a7f99222eb03bf296260fac7b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/882fe79977fce2cc7c289b0da1721c2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6581916f5a65edfea257c804efee007e.png)
(1)在同一直角坐标系中,求平面直角坐标系中的伸缩变换
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe4f007b1ceaccfff1d659f6f8592c4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78847dd23bb54d5d960016e6beeb5713.png)
(2)在同一直角坐标系中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/882fe79977fce2cc7c289b0da1721c2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a21e44af368f3336354704d92609f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a21e44af368f3336354704d92609f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/150a135bbd528daf3f19a58a621a57c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/793b3bfc799e7cd6a795324ca02aaa23.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e5e61804ce550636a0354e0a78a22d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e5e61804ce550636a0354e0a78a22d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e5e61804ce550636a0354e0a78a22d.png)
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2卷引用:上海市七宝中学2023届高三上学期元月模拟数学试题
9 . 在平面直角坐标系中,曲线C1的方程为
,曲线C2的参数方程为
(t为参数),直线l过原点O且与曲线C1交于A、B两点,点P在曲线C2上且OP⊥AB.以O为极点,x轴正半轴为极轴建立极坐标系.
(1)写出曲线C1的极坐标方程并证明
为常数;
(2)若直线l平分曲线C1,求△PAB的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f99b71bea536a435f558d319e9af1eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40364f6c7aa7137aef5a54a18358620e.png)
(1)写出曲线C1的极坐标方程并证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff24c95fe581befd56c3bcc70e88b726.png)
(2)若直线l平分曲线C1,求△PAB的面积.
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8卷引用:四川省德阳市2023届高三第一次诊断考试数学(文)试题
四川省德阳市2023届高三第一次诊断考试数学(文)试题(已下线)四川省德阳市高中2022-2023学年高三上学期第一次诊断考试文科数学试题四川省德阳市高中2022-2023学年高三上学期第一次诊断考试理科数学试题(已下线)专题20坐标系与参数方程四川省泸州市泸县第四中学2024届高三一模数学(文)试题2024届四川省泸州市泸县第五中学高三一模文科数学试题2024届四川省泸州市泸县第五中学高三一模理科数学试题四川省泸州市泸县第四中学2024届高三一模数学(理)试题
10 . 直线l过点
,倾斜角为
.
(1)以平面直角坐标系的原点O为极点,x轴的正半轴为极轴建立极坐标系.过O作l的垂线,垂足为B,求点B的极坐标
;
(2)与曲线
(t为参数)交于
两点,证明:
成等比数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6147cff1a5f9043fbb81f25d836f2e6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af9955b5aebb73cd84447e8541f901ac.png)
(1)以平面直角坐标系的原点O为极点,x轴的正半轴为极轴建立极坐标系.过O作l的垂线,垂足为B,求点B的极坐标
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e005472741b350a2d8c732c062ebfe9e.png)
(2)与曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47a7e0edcc0927bf5a009c027faee413.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f44755c5fee4b90266eac73ad47a128.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a3f786ffb72b3739898186281ddd6a3.png)
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2022-08-27更新
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4卷引用:东北三省四市教研联合体2022届高三下学期模拟试卷(二)文科数学试题