解题方法
1 . 如图矩形
中,
,沿对角线
将
折起,使点A折到点P位置,若
,三棱锥
的外接球表面积为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/9/56684e59-852e-4646-9799-ae1ca928dce5.png?resizew=173)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/9/39359f77-1463-4b17-9518-73992753a315.png?resizew=199)
(1)求证:平面
平面
;
(2)求平面
与平面
夹角的余弦值;
(3)M为
的中点,点N在
边界及内部运动,若直线
与直线
与平面
所成角相等,求点N轨迹的长度.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fbb42439079fa563100decbad833e10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08a9ec3b527947cad9caa4537e0cb7e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4a39dce3f1e36dbe01293c309816968.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/9/56684e59-852e-4646-9799-ae1ca928dce5.png?resizew=173)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/9/39359f77-1463-4b17-9518-73992753a315.png?resizew=199)
(1)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f04c222223dae9ef27d4c132534d9848.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b60870baa5e3fbc33a749aa5f0a94be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
(3)M为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661ff55b5ebbadfb600989af3cfce2fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8ffe24cf9f327aeb241225ab15ab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
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解题方法
2 . 如图所示,由半椭圆
和两个半圆
、
组成曲线
,其中点
依次为
的左、右顶点,点
为
的下顶点,点
依次为
的左、右焦点.若点
分别为曲线
的圆心.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/15/f030a695-e70d-49de-899f-fb7a8254fd58.png?resizew=201)
(1)求
的方程;
(2)若点
分别在
上运动,求
的最大值,并求出此时点
的坐标;
(3)若点
在曲线
上运动,点
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b58682ebbe3fb045b531d941a4f6e1ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc1cbad6f9e495834f0920e63c0351f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f026d5938eb00a18e9933312b9008a0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/831e26f0b6438b1096e75e608337fc80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/663368000ac90f582d12675aa2d1e832.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c38928a92bc4b44ed3c9b89769f5372.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c38928a92bc4b44ed3c9b89769f5372.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f3a48942090e536757aa060dada78e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/15/f030a695-e70d-49de-899f-fb7a8254fd58.png?resizew=201)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5b1b15a4605fce993cb13aefbf40360.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f3a48942090e536757aa060dada78e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64ac6820c1dce7bb381d972d3ad68403.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5b1b15a4605fce993cb13aefbf40360.png)
(3)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/831e26f0b6438b1096e75e608337fc80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fef26f14e6a51c897b4dd93c72ad133.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d3ac506d3db8a365fa3f136a589fbeb.png)
您最近一年使用:0次
2022-12-15更新
|
797次组卷
|
2卷引用:上海市嘉定区2023届高三上学期一模数学试题
名校
解题方法
3 . 已知圆
,定点
.
(1)过点
作圆
的切线,切点是A,若线段
长为
,求圆
的标准方程;
(2)过点
且斜率为1的直线
,若圆
上有且仅有4个点到
的距离为1,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd06dbe851fd8c6a53e275a41603c8a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70f6c19987f273b62f5bbe6554b9efe5.png)
(1)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b66a5b7813e902306477f91f9f4084cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ce0249a3ff99c083fa4421877549db1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过点
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2022-11-28更新
|
828次组卷
|
5卷引用:湖北省襄阳市老河口市第一中学2022-2023学年高二上学期元月月考数学试题
4 . 求满足下列条件的曲线方程
(1)已知直线
与圆心在原点的圆C相切,求圆C的方程.
(2)已知椭圆的两个焦点分别是
和
,并且经过点
求椭圆标准方程.
(3)求平行于直线
,且与它的距离为
的直线方程.
(1)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2db654cadf6b5483e8aec45e9bca7a00.png)
(2)已知椭圆的两个焦点分别是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccc8350b12974ffc8d06fce36d158f02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a812a9b58ccba331cfd21d244329af01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301a95724aeecaa563b18c4499df7e22.png)
(3)求平行于直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c6d3c99b7004603ba9ea9c341b8b3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95bacae35b6e16a0a33c2bdc6bc07df7.png)
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2022-11-28更新
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2卷引用:湖北省襄阳市老河口市第一中学2022-2023学年高二上学期元月月考数学试题
名校
5 . 周口市沙河湾湿地公园内有一直角梯形区域
,
,
,
.相关部门欲在
,
两处各建一个景点,将
边建成人行步道(人行步道的宽度忽略不计).
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/26/b291ebd2-9542-432c-8744-103a6710b6b3.png?resizew=112)
(1)若分别以
,
为圆心的两个圆都与直线
相切,且这两个圆外切,求
,
两点之间的距离;
(2)若
,今欲在人行步道(线段
)上设一观景台
,已知观景台
在过
,
两点的圆与直线
相切的切点处时,有最佳观赏和拍摄的效果,问观景台
设在何处时,观赏和拍摄的效果最佳?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4adf90a8c2b29334cdc5aa5b554991f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdb2dd10731b99c0f4f89ee957f8a239.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a48324d6d5c33285ea76b67b4f0a5018.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/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/26/b291ebd2-9542-432c-8744-103a6710b6b3.png?resizew=112)
(1)若分别以
![](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/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d390b4acff7221e22e7f84358cab17dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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2022-11-26更新
|
197次组卷
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2卷引用:河南省沁阳市永威学校2023-2024学年高二上学期第二次月考数学试题
解题方法
6 . 已知圆C,圆
,圆
这三圆有一条公共弦.
(1)当圆C的面积最小时,求圆C的标准方程;
(2)在(1)的条件下,直线l满足:
(ⅰ)与直线
平行;
(ⅱ)与圆C相切.
若直线l与圆
分别交于A,B两点,与圆
分别交于D,E两点,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d6ac185918af575c60aa91b070ee612.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88002107328c7669fe3cb5fdeb14efd8.png)
(1)当圆C的面积最小时,求圆C的标准方程;
(2)在(1)的条件下,直线l满足:
(ⅰ)与直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89646b149f703d83ea142d297bfffb46.png)
(ⅱ)与圆C相切.
若直线l与圆
![](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/991444dcdadfebbef3208c68467c13e2.png)
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解题方法
7 . 已知圆C经过
,
两点.
(1)当
时,圆C与x轴相切,求此时圆C的方程;
(2)如果AB是圆C的直径,证明:无论a取何正实数,圆C恒经过除A外的另一个定点,求出这个定点坐标.
(3)已知点A关于直线
的对称点
也在圆C上,且过点B的直线l与两坐标轴分别交于不同两点M和N,当圆C的面积最小时,试求
的最小值;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/383f12cb70ca55eba4ff012771dbfa9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ffa2d3ca8ca3dad75e14272e82155ba.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
(2)如果AB是圆C的直径,证明:无论a取何正实数,圆C恒经过除A外的另一个定点,求出这个定点坐标.
(3)已知点A关于直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e235d7dd12f948f5ffb2e5afddc95612.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/975d6f7561fc2bce5f4c94a33b3c0ccf.png)
您最近一年使用:0次
2022-09-11更新
|
1043次组卷
|
4卷引用:江苏省无锡市辅仁高级中学2023-2024学年高二上学期10月月考数学试题
名校
8 . “跳台滑雪”是冬奥会中的一个比赛项目,俗称“勇敢者的游戏”,观赏性和挑战性极强.如图:一个运动员从起滑门点
出发,沿着助滑道曲线
滑到台端点
起跳,然后在空中沿抛物线
飞行一段时间后在点
着陆,线段
的长度称作运动员的飞行距离,计入最终成绩.已知
在区间
上的最大值为
,最小值为
.
![](https://img.xkw.com/dksih/QBM/2022/4/26/2966573015228416/2967895167041536/STEM/8ed17a7d-801c-4717-8dae-c1014befe9b0.png?resizew=244)
(1)求实数
,
的值及助滑道曲线
的长度.
(2)若运动员某次比赛中着陆点
与起滑门点
的高度差为120米,求他的飞行距离(精确到米,
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19b62fb7a9f329186d2561ab9a2da883.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efefd0f7ee12b6938f7442904ef51dd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68196c2c072d591a129a259fb377f8e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/787c68fc75db2e618c66574911546b93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f86b86a6ccdf2c077d195db7bc380ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/545d8e23d43c373fcbf595ade07f8c4b.png)
![](https://img.xkw.com/dksih/QBM/2022/4/26/2966573015228416/2967895167041536/STEM/8ed17a7d-801c-4717-8dae-c1014befe9b0.png?resizew=244)
(1)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(2)若运动员某次比赛中着陆点
![](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/368763128d1ad0ffad5d859fef834d0a.png)
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2022-04-28更新
|
1297次组卷
|
9卷引用:专题13 函数模型及其应用
(已下线)专题13 函数模型及其应用(已下线)专题13 函数模型及其应用-2四川省绵阳市江油市太白中学2022-2023学年高二下学期期中数学文科试题(已下线)第08讲 圆的方程(3大考点九种解题方法)(2)上海市上海交通大学附属中学2022届高三下学期期中数学试题福建省永春第一中学2021-2022学年高二下学期期末考试数学试题四川省达州市宣汉县土黄中学2022-2023学年高二上学期第一次测试数学试题(已下线)专题01 集合与不等式必考题型分类训练-4辽宁省实验中学2022-2023学年度高三上学期12月教学质量检测数学试题
名校
9 . 已知点
,
,点A满足
,点A的轨迹为曲线C.
(1)求曲线C的方程;
(2)若直线
与双曲线:
交于M,N两点,且
(O为坐标原点),求点A到直线l距离的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2e56502edba9511c1ce84ecbf6d2e99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8d89e9d012179b974e2c6bb481fa07b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17b8dfe216ef4fed64026f3bf694cc23.png)
(1)求曲线C的方程;
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fc5bd66dd6d5e09ff0893a938aed56e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14f8ae565db8a7d4b95fdd85e28afa2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4489ea382b7a51cd5fc8c5ae585a3ec4.png)
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2022-04-08更新
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1971次组卷
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5卷引用:陕西师范大学附属中学渭北中学2022-2023学年高二下学期5月月考文科数学试题
解题方法
10 . 在平面直角坐标系中,
为坐标原点,曲线
上的点都在
轴及其右侧,且曲线
上的任一点
到
轴的距离比它到圆
的圆心的距离小1.
(1)求曲线
的方程;
(2)已知过点
的直线
交曲线
于点
,若
,求
面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e27fdece4d9158ab6454430b0032c599.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.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/ccfd4a64854b9dbc89aee3fd2b094600.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
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