名校
解题方法
1 . 类似平面解析几何中的曲线与方程,在空间直角坐标系中,可以定义曲面(含平面)
的方程,若曲面
和三元方程
之间满足:①曲面
上任意一点的坐标均为三元方程
的解;②以三元方程
的任意解
为坐标的点均在曲面
上,则称曲面
的方程为
,方程
的曲面为
.已知曲面
的方程为
.
过曲面
上一点
,以
为方向向量,求证:直线
在曲面
上(即
上任意一点均在曲面
上);
(2)已知曲面
可视为平面
中某双曲线的一支绕
轴旋转一周所得的旋转面;同时,过曲面
上任意一点,有且仅有两条直线,使得它们均在曲面
上.设直线
在曲面
上,且过点
,求异面直线
与
所成角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1277d620abbfa26fb39600e53e606d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1277d620abbfa26fb39600e53e606d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1277d620abbfa26fb39600e53e606d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fa13f27f92b55bd7ecd9d750f98ae99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1277d620abbfa26fb39600e53e606d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1277d620abbfa26fb39600e53e606d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eedd047376c4cf1b9992cd8e4fe20df2.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/33d7a5b312e3d789a1070000315d63b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0b4e60b50889c152c56fcf2d6ea35bc.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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知曲面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7d96461d2b3421aed548b754637ca8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.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/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44ddad9072eb1393ce15ca0627c941b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
您最近一年使用:0次
名校
解题方法
2 . 已知点
,动点
到直线l:
的距离为d,且
,记S的轨迹为曲线C.
(1)求曲线C的方程;
(2)若
,
分别为曲线C的左、右顶点,M,N两点在直线
上,且
.连接
,
分别与C交于点P,Q,求证:直线PQ过定点,并求出定点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14436636ec6a7aec09cb63cecf6e970d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ee683bfdad9234a4aa810135e0d00e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fe5ef58ae020856fba17dfbee68c321.png)
(1)求曲线C的方程;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39cc033406da2cdd342308972c6701f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eb443ea94c7aca4b54e4808b4560525.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9399c9a2a31b0e3165aea2d6ccc4f7c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b06b75fb4e379ff3b99e68f40136cad.png)
您最近一年使用:0次
名校
解题方法
3 . 已知椭圆
的上、下顶点分别为A,B,O为椭圆的中心,D是线段OB的中点.直线
,动点T到直线m的距离与T到点
的距离相等.设动点T的轨迹为
.
(1)求
的方程;
(2)过点D作斜率为
的直线l,交
于M,N,直线
分别交
于P,Q两点(P,Q均不同于点A),设直线
的斜率为
,求证:
是定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a0ad41722598d9fd00138a0f781b5df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28c2debb1285d2bb3b7907bece265c01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0436039bcb24dab81c2664575dbad83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(2)过点D作斜率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e2fd6df03fd6e945e2c1cb44bca014b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d42cb68c5c877a455ba7ac0a6b6a651.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2a3f348a942d468f0d77c0dfbb41d87.png)
您最近一年使用:0次
4 . 已知
,动点
满足
与
的斜率之积为定值
.
(1)求动点
的轨迹
的方程;
(2)过点
的直线
与曲线
交于
两点,且
均在
轴右侧,过点
作直线
的垂线,垂足为
.
(i)求证:直线
过定点;
(ii)求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9dc1fd4a7d898e9d7307255cc21678e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ad3a1ea6790177130e16c2124984087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccedd9e7b7f0e54838f58d984519cec6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
(1)求动点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65368687df4d7e3b9304e85ec4de354c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.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/132a241ff20784c763b27ab8323a50fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(i)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
(ii)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b9f0296c53918018745f4e3906e2dd8.png)
您最近一年使用:0次
名校
解题方法
5 . 如图,四棱锥中,
底面
,四边形
中,
,
.
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78a3fd5284e160896f07ce367645fd04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
(2)若平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64f1145c162038df3c7184d9201c628e.png)
(ⅰ)求线段的长;
(ⅱ)设为
内(含边界)的一点,且
,求满足条件的所有点
组成的轨迹的长度.
您最近一年使用:0次
2024-01-17更新
|
1848次组卷
|
4卷引用:重庆市主城区2024届高三上学期第一次学业质量检测数学试题
重庆市主城区2024届高三上学期第一次学业质量检测数学试题福建省永春一中、培元中学、石光中学、季延中学2024届高三下学期第二次联合考试数学试题(已下线)第三章 空间轨迹问题 专题三 立体几何轨迹长度问题 微点2 立体几何轨迹长度问题综合训练【培优版】(已下线)专题04 立体几何
6 . 已知
,曲线
上任意一点到点
的距离是到直线
的距离的两倍.
(1)求曲线
的方程;
(2)已知曲线
的左顶点为
,直线
过点
且与曲线
在第一、四象限分别交于
,
两点,直线
、
分别与直线
交于
,
两点,
为
的中点.
(i)证明:
;
(ii)记
,
,
的面积分别为
,
,
,则
是否为定值?若是,求出这个定值;若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db5660be1b6e0d3d1cab756d6f8f5855.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b650820d7bed48ed67a2869ad8c65ff1.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知曲线
![](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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b650820d7bed48ed67a2869ad8c65ff1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35d58f9019097bd05037aefd5c322916.png)
(i)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b85d79eabe4ce65b09f16a202c6b017.png)
(ii)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba08ef7e19164e78dc0dc08fe4395d37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/135361ee2e038ca0dd890b701ac1ef5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/740eea87967a331d7da821592a639076.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cc80d8bbc5fec873686719e78bd518d.png)
您最近一年使用:0次
名校
解题方法
7 . 已知点
在曲线
上,
为坐标原点,若点
满足
,记动点
的轨迹为
.
(1)求
的方程;
(2)设
是上
的两个动点,且以
为直径的圆经过点
,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c99e8488f37ecf147b0bf7663b66f052.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6690b42f6997550f086e4a4cb5a145d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c5eb2192f8c5a804d19afb8d9157ce2.png)
您最近一年使用:0次
名校
解题方法
8 . 已知抛物线
的焦点为
是抛物线上一点且三角形MOF的面积为
(其中O为坐标原点),不过点M的直线l与抛物线C交于P,Q两点,且以PQ为直径的圆经过点M,过点M作
交PQ于点N.
(1)求抛物线C的方程;
(2)求证直线PQ恒过定点,并求出点N的轨迹方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c68c78768c0b779356a139a4dad1e7d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ca8b26c3ad6d892590290a2304126bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8c957b71beace01e35905eccbca90d.png)
(1)求抛物线C的方程;
(2)求证直线PQ恒过定点,并求出点N的轨迹方程.
您最近一年使用:0次
2021-10-05更新
|
477次组卷
|
4卷引用:重庆市乌江新高考协作体2023-2024学年高二上学期期末学业质量联合调研抽测数学试题
重庆市乌江新高考协作体2023-2024学年高二上学期期末学业质量联合调研抽测数学试题陕西省咸阳市2021-2022学年高三上学期开学摸底考试文科数学试题(已下线)一轮复习大题专练69—抛物线3(定点问题2)—2022届高三数学一轮复习贵州省兴义市第八中学2024届高三上学期第一次月考数学试题