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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解题方法
2 . 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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解题方法
3 . 已知F为抛物线
(t为参数)的焦点,过F作两条互相垂直的直线
,直线
与C交于A,B两点,直线
与C交于D,E两点,则
的最小值为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/512d63e6d79877f4407a6a8f68dce3e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192aeb9fe486f44a27d837274daf8657.png)
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2024-02-13更新
|
326次组卷
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3卷引用:陕西省2024届高三教学质量检测(一)文科数学试题
陕西省2024届高三教学质量检测(一)文科数学试题陕西省2024届高三教学质量检测(一)理科数学试题(已下线)第2章 圆锥曲线 单元综合检测(难点)-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)
名校
解题方法
4 . 已知实数
,
满足
,则代数式
的最大值为______ .
![](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/2c9fbdbc58b8268344cab615b33986c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/285bb8dac8bdbbda922db848827578bf.png)
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2023-12-02更新
|
505次组卷
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2卷引用:湖南省浏阳市2023-2024学年高二上学期期末质量监测数学试卷
5 . 在直角坐标系
中,点
,直线
.设动点
到
的距离为
,且
.以点
为极点,
轴正半轴(
点右侧)为极轴,建立极坐标系.
(1)求
轨迹
的极坐标方程;
(2)直线
为参数),与
交于
、
两点,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8748dc55e2f45bc37fc4d84d7310f79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e2977eea43a781e06d93e04a395a309.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/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0667fd53392abfa58d360e78814eaf51.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/5963abe8f421bd99a2aaa94831a951e9.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c33a784ffdf8c129f217c4e572df097a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d44e8bc37ed03f44470762748a8f942a.png)
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6 . 在直角坐标系
中,曲线
的参数方程为
(
为参数),以坐标原点
为极点,
轴的正半轴为极轴建立极坐标系,直线
的极坐标方程是
.
(1)求曲线
的普通方程和直线
的直角坐标方程;
(2)设直线
与曲线
交于
,
两点,
,求
的值.
![](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/f3d56ae53ed70ceb752eecb91a210084.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.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/5e5ac03ca0fa0b07d788ccee0392206b.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/c5db41a1f31d6baee7c69990811edb9f.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/22840186db0afc0e2b2e8915ce79b998.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83c1a099b4673fe2ae0f084beae1d0b2.png)
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7 . 已知曲线
的参数方程为
(
为参数),直线
过点
.
(1)求曲线
的普通方程;
(2)若直线
与曲线
交于
,
两点,且
,求直线
的倾斜角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a4c4c9cf8f0bd3f84fdc5bae0aa65b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1b6f209d1a805437046ca6ef79dd.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若直线
![](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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffea71f4a401758f650444e16c10d017.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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2023-04-10更新
|
1243次组卷
|
6卷引用:四川省成都市第七中学2024届高三下学期4月分推考试数学(理科)试卷
名校
8 . 在直角坐标系xOy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
的极坐标方程为
.
(1)写出
的直角坐标方程;
(2)已知点
,若l与C交于A,B两点,且
,求m的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dadca2f308ede855eb1e25955ade087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf0bde847e90666da28bec644998cf22.png)
(1)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce171aaebc875b97e60a64bf4f2c59f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dc6645fb7762ce2d6ae18439296c65f.png)
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2023-03-02更新
|
2311次组卷
|
8卷引用:2024年高考全国甲卷数学(理)真题平行卷(提升)
(已下线)2024年高考全国甲卷数学(理)真题平行卷(提升)四川省泸州市2023届高三下学期第二次教学质量诊断性考试数学(文科)试题四川省泸州市2023届高三第二次教学质量诊断性考试数学(理科)试题四川省成都市实验外国语学校2023届高三第五次质量检测数学文科试题四川省成都市实验外国语学校2023届高三第五次质量检测数学理科试题四川省成都市简阳市阳安中学2022-2023学年高二下学期5月月考数学(理)试题(已下线)2023年高考全国甲卷数学(文)真题变式题21-23(已下线)2023年高考全国甲卷数学(理)真题变式题21-23
名校
解题方法
9 . “曼哈顿距离”是由赫尔曼·闵可夫斯基所创的词汇,是一种使用在几何度量空间的几何学用语.在平面直角坐标系中,点
,
的曼哈顿距离为
.若点
,Q是圆
上任意一点,则
的取值范围为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724d316a295242846ae0f70a18e1e659.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac51d5b0a1b6d229ca8e73300843e3b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/845c73ecca715e281ee13defac47f1e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dacabbd6406acbb4afb048e9cfa1bcf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34f36e22874cc6dd08e960ecdcca58a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/135b08696706de37e1eab5f59697674c.png)
A.![]() | B.![]() | C.![]() | D.![]() |
您最近一年使用:0次
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解题方法
10 . 实数
满足
,则
的最小值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d10449bc77d692a7270e0f20a68cdf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d63471465a852901b6aaf6b7fc4d042.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e159c8203722a3c50784f7f6e65911e2.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
您最近一年使用:0次
2022-07-05更新
|
1087次组卷
|
3卷引用:河南省郑州市郑州外国语学校2024届高三上学期适应性训练数学试题