2024·全国·模拟预测
名校
解题方法
1 . 已知直线
和椭圆
.
(1)证明:
与
恒有两个交点;
(2)若
为
与
的两个交点,过原点且垂直于
的直线交
于
两点,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b648c913b1b66abb9cd526dd8a7b2ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0b0ae6048bf94ccd5a3b4a8aafd81e.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ce9db8bf6dff7b5b1feb849f5532f22.png)
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名校
解题方法
2 . 已知抛物线
的准线过椭圆E:
的左焦点,且椭圆E的上顶点与两个焦点构成一个正三角形.
(1)求椭圆E的方程;
(2)直线
交椭圆E于A,B两点,点P在线段
上移动,连接
交椭圆于M,N两点,过P作
的垂线交x轴于Q,求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
(1)求椭圆E的方程;
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41a5ff72ba4e9d01ecf0c0fe07a48058.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed283a253b61df01f2a1cdc0cd8003f3.png)
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3 . 已知抛物线
的焦点
到准线
的距离为6.
(1)求抛物线
的方程;
(2)已知点
,
是
上的两点,
是抛物线
上一动点,原点到直线PA,PB的距离均为3,求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ba379e55fcf7ceee03de7d43001eeb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c8e73eb063f711a944a9843bdd55ad4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1858507c21102b33fadc14d42aa38441.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
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2024-04-10更新
|
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4 . 平面内,过点
和
的两条直线交于点P,且直线
和直线
的斜率之积为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/602baac86c2b1668ecdfadc8a5948885.png)
(1)求点P的轨迹C的方程;
(2)过点
的直线
交
点的轨迹C于
、
两点,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/913f78382630e50543e5f7192cae3ed3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c850811ba59a05e945a665196539a048.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/602baac86c2b1668ecdfadc8a5948885.png)
(1)求点P的轨迹C的方程;
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2a29ba49963134a7232fa8574105fc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04f633fe79fadde8fc62745fe1b3624d.png)
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解题方法
5 . 在棱长为
的正方体
中,
、
分别是棱
、
的中点,动点
满足
,
,下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0951bf5509da7ec1ef0b8e8e49c6952.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2187ee1ea3b7e47a6283314322e5decf.png)
A.当![]() ![]() ![]() |
B.当![]() ![]() ![]() ![]() |
C.当![]() ![]() ![]() ![]() |
D.设直线![]() ![]() ![]() ![]() |
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6 . 已知
是双曲线
的右焦点,
是
左支上一点,
,当
周长最小时,该三角形的面积为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99022e61931b203ad3763bde2a1e3852.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e29672dfdea3d040dfd7973b3d4a56b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ceff3844281849df3e37a2e56e110549.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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|
438次组卷
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3卷引用:吉林省长春吉大附中实验学校2023-2024学年高二上学期期中考试试题
名校
解题方法
7 . 已知双曲线
的左、右顶点分别为A,B,P是C上任意一点,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6030294837c740b4fe4bb00162137e38.png)
A.C的渐近线方程为![]() |
B.若直线![]() ![]() |
C.点P到C的两条渐近线的距离之积为![]() |
D.当点P与A,B两点不重合时,直线PA,PB的斜率之积为2 |
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名校
解题方法
8 . 如图,在棱长为2的正方体
中,
,
分别是棱
,
的中点,点
在
上,点
在
上,且
,点
在线段
上运动,下列说法正确的有( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/9/9800205b-1dce-4d32-994a-068234a2f264.png?resizew=165)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.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/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/394c5d2f55221975503be8aa18022480.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f6f93171329d508d491143b9d71f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f805768a5ffaf8bdfa4bc3b680aafdc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/9/9800205b-1dce-4d32-994a-068234a2f264.png?resizew=165)
A.当点![]() ![]() ![]() ![]() |
B.直线![]() ![]() ![]() |
C.存在点![]() ![]() |
D.![]() ![]() |
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2023-10-25更新
|
1028次组卷
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7卷引用:吉林省松原市前郭尔罗斯蒙古族自治县蒙古族中学2023-2024学年高二上学期期中数学试题
吉林省松原市前郭尔罗斯蒙古族自治县蒙古族中学2023-2024学年高二上学期期中数学试题广东省广州天省实验学校2023-2024学年高二上学期期中数学试题四川省江油市太白中学2023-2024学年高二上学期期中数学试题山东省淄博市实验中学、齐盛高级中学2023-2024学年高二上学期期中考试数学试卷浙江省绍兴蕺山外国语学校2023-2024学年高二上学期9月检测数学试题(已下线)广东省佛山市南海区桂城中学2023-2024学年高二上学期11月月考数学试题(已下线)黄金卷01
名校
解题方法
9 . 在直三棱柱
中,底面
为等腰直角三角形,且满足
,点
满足
,其中
,
,则下列说法正确的是( )
![](https://img.xkw.com/dksih/QBM/2023/10/19/3349210988781568/3349896098291712/STEM/33ba68ac8ce3410fbc9dfc847077b0c3.png?resizew=169)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2af2626608f61a4cfbb86494bd6df0e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac6b545127bd51036a5a7b0d3cd5b320.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cee72261f6901e62dfd0ffe547406544.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34e2e01346f60857ff635bb766802e57.png)
![](https://img.xkw.com/dksih/QBM/2023/10/19/3349210988781568/3349896098291712/STEM/33ba68ac8ce3410fbc9dfc847077b0c3.png?resizew=169)
A.当![]() ![]() ![]() ![]() |
B.当![]() ![]() |
C.当![]() ![]() ![]() |
D.当![]() ![]() ![]() ![]() |
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2023-10-20更新
|
966次组卷
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5卷引用:吉林省长春市长春外国语学校2023-2024学年高三上学期期中数学试题
吉林省长春市长春外国语学校2023-2024学年高三上学期期中数学试题辽宁省大连市第八中学2023-2024学年高二上学期期中考试数学试题福建省福州市闽侯县第一中学2023-2024学年高二上学期10月月考数学试题江西省上饶市广信中学2023-2024学年高二上学期11月月考数学试题(已下线)第一章 点线面位置关系 专题二 空间垂直关系的判定与证明 微点3 直线与平面垂直的判定与证明【基础版】
10 . 已知点
在椭圆
上,设点
为
的短轴的上、下顶点,点
是椭圆上任意一点,且
,
的斜率之积为
.
(1)求
的方程;
(2)过
的两焦点
、
作两条相互平行的直线
,
交
于
,
和
,
,求四边形
面积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd243fab0af865af67a2ab817e909cf5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e98c4b3f3fe826e124ca7d199d4ca4b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/817a419430d9951cbdb89b657b21bcf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d78fd95f89dec2d373fa57f02acd739f.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.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/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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17cb7a12bb8797615303fc20adb9f988.png)
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2023-10-09更新
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1240次组卷
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5卷引用:吉林省长春市南关区长春市实验中学2023-2024学年高二上学期期中数学试题
吉林省长春市南关区长春市实验中学2023-2024学年高二上学期期中数学试题广东省佛山市顺德区华侨中学2024届高三港澳班上学期期中数学试题四川省眉山市眉山冠城七中实验学校2023-2024学年高二上学期期中数学试题重庆市第八中学校2023-2024学年高二上学期第一次月考数学试题(已下线)专题08 椭圆双曲线综合大题(9题型)-【巅峰课堂】2023-2024学年高二数学上学期期中期末复习讲练测(人教A版2019选择性必修第一册)