1 . 数学家加斯帕尔·蒙日在研究圆锥曲线时发现:椭圆
任意两条互相垂直的切线的交点都在以原点O为圆心,
为半径的圆上,这个圆被称为该椭圆的蒙日圆.已知椭圆C:
可以与边长为
的正方形的四条边均相切,它的左、右顶点分别为A,B,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da001dad7941e6c9858637d7b62cec59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec92768b052f6ceb3efc112b7e48625.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b10e8abf8690e4b129466ddb918bcc94.png)
A.![]() |
B.若矩形的四条边均与椭圆C相切,则该矩形面积的最大值为12 |
C.椭圆C的蒙日圆上存在两个点M满足![]() |
D.若椭圆C的切线与C的蒙日圆交于E,F两点,且直线OE,OF的斜率都存在,记为![]() ![]() ![]() |
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解题方法
2 . 已知点P在圆
上,过点P作x轴的垂线段
,D为垂足,Q为线段
的中点,当点P在圆上运动时,点Q的轨迹为Γ.
(1)求Γ的方程;
(2)设
,
,过点
作直线与Γ交于不同的两点M,N(异于A,B),直线
,
的交点为G.
(ⅰ)证明:点G在一条平行于x轴的直线上;
(ⅱ)设直线
,
交点为H,试问:
与
的面积之积是否为定值?若是,求出该定值;若不是,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5f5d967ad135991b6075ee45df55643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
(1)求Γ的方程;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/383f12cb70ca55eba4ff012771dbfa9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3544997cc034ed882c0d0a3bdbf5f957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0e1ba8ef888dfe9a639dddd38d6d603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
(ⅰ)证明:点G在一条平行于x轴的直线上;
(ⅱ)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3b785ebbf5889849e872f461669f71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f584dfa75ec20e4cba4216998b454dd.png)
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2024·山东·模拟预测
解题方法
3 . 设异面直线
与
所成的角为
,公垂线段为
,且
,
、
分别直线m、n上的动点,且
,
为线段
中点,建立适当的平面直角坐标系可确定点
的轨迹方程
.
(1)请根据自己建立的平面直角坐标系求出
.
(2)
为
的任意内接三角形,点
为
的外心,若直线
的斜率存在,分别为
,
,
,
,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbe3080ed8cb377910703dd956207fa9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1da895d8bd043625a0839128252130d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28b1ba4307cfde9b424d468bfcdf6c5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0e4a2447b62462f97d8dcf59fdff728.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0f9b255a8da2b914a84a848e0ec499d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(1)请根据自己建立的平面直角坐标系求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80792ba7adda47f55640c29312ccc496.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbf434334b09cc0fdd4e86e84e6ceb00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3307e11f7e6896e32aa510bbed949ac6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dee82d6e88e4e5e88fdf8896815ca678.png)
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解题方法
4 . 在平面直角坐标系
中,重新定义两点
之间的“距离”为
,我们把到两定点
的“距离”之和为常数
的点的轨迹叫“椭圆”.
(1)求“椭圆”的方程;
(2)根据“椭圆”的方程,研究“椭圆”的范围、对称性,并说明理由;
(3)设
,作出“椭圆”的图形,设此“椭圆”的外接椭圆为
的左顶点为
,过
作直线交
于
两点,
的外心为
,求证:直线
与
的斜率之积为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e3a1467ecf286e3cadaf5aa006606f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9859b2a9747b7a9da0b87624231e5a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de32f743ea0cf45f9822dd603be212d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd95e24f73f48167eb193951bd4fa7fb.png)
(1)求“椭圆”的方程;
(2)根据“椭圆”的方程,研究“椭圆”的范围、对称性,并说明理由;
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/715c3978c454777672e14a72298317a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0dd7df0a96857b265fbbf745873ace9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98013a5042685a1db94249e70c62c09a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f0009063fe00277645aff1be6e32471.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
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2024-03-22更新
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3卷引用:山东省菏泽市第三中学2024届高三下学期3月月考数学试题
5 . 已知椭圆
:
的左右焦点分别为
,
,且点
是直线
上任意一点,过点
作
的两条切线
,
,切点分别为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cae00bdc6f8b564b6b15b32572c848b.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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.png)
![](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/8e5c62f22d7afc5627fcb86599faa8e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
A.![]() | B.A,![]() ![]() |
C.A,![]() | D.![]() |
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6 . 如图,已知圆
,圆心是点T,点G是圆T上的动点,点H的坐标为
,线段CH的垂直平分线交线段TC于点R,记动点R的轨迹为曲线E.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/17/7f184f36-d38a-4a16-a68d-696afedbb746.png?resizew=152)
(1)求曲线E的方程;
(2)过点H作一条直线与曲线E相交于A,B两点,与y轴相交于点C,若
,
,试探究
是否为定值?若是,求出该定值;若不是,请说明理由;
(3)过点
作两条直线MP,MQ,分别交曲线E于P,Q两点,使得
.且
,点D为垂足,证明:存在定点F,使得
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7ebdc37c4828abb80c0f3bffb9e54e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5510fbff7fdcc083ef172d4b401c2229.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/17/7f184f36-d38a-4a16-a68d-696afedbb746.png?resizew=152)
(1)求曲线E的方程;
(2)过点H作一条直线与曲线E相交于A,B两点,与y轴相交于点C,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9545ecde4068ea16914263cddb3e8f60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1cff2d267cabb5318d29e319fe65c90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/febf7413b35cf2889fdb57a6b519087c.png)
(3)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52b385adf402cd17bddace02fdb32030.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85ebcef3ed51c751de97752bbf40e87c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41b14463c252e686a3cd0a8c4e33c02b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0165236f973be829fd3dbabe9507243.png)
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2023·全国·模拟预测
名校
解题方法
7 . 已知椭圆C:
的两焦点分别为
,并且经过点
.
(1)求椭圆C的方程;
(2)过
的直线交椭圆C于A,B两点,设直线
与C的另一个交点分别为M,N,记直线AB,MN的倾斜角分别为
,当
取得最大值时,求直线AB的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5439f5ff9bd5deec0f0ef35c6f605b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a34dda3fc71497fe1a7ef2f3d2c2a953.png)
(1)求椭圆C的方程;
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0068ae8bb3fb1c75a3843e590fb30607.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd927b4b5a7875528c1b54aa4bb8b2dd.png)
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2024-01-02更新
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6卷引用:山东省菏泽市2024届高三上学期期末考试数学试题(B)
山东省菏泽市2024届高三上学期期末考试数学试题(B)(已下线)2024年全国高考名校名师联席命制型数学信息卷(七)(已下线)2024年全国高考名校名师联席命制数学(理)信息卷(四)(已下线)2024年全国高考名校名师联席命制数学(文)信息卷(三)江西省上饶市玉山县第二中学2023-2024学年高二上学期12月月考数学试题四川省成都市石室中学2023-2024学年高二上学期期末综合复习数学试题(一)
8 . 在直角坐标平面内,已知
,动点
满足条件:直线
与直线
的斜率之积等于
,记动点
的轨迹为曲线
.
(1)求曲线
的方程;
(2)过点
作直线
交
于
两点(与
不重合),直线
与
的交点
是否在一条定直线上?若是,求出这条定直线的方程;若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3c9967968279271d8cf1f9444c0ac1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/158b045c6172c4178d7aa52083e1489f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8860a9c949f912f01dfc58d002d387cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
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2023-12-15更新
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4卷引用:山东省菏泽市鄄城县第一中学2023-2024学年高二上学期12月月考数学试题
名校
解题方法
9 . 已知椭圆
的离心率为
,短轴长为2.
(2)如图,已知A,B,C为椭圆E上三个不同的点,原点O为
的重心;
①如果直线AB,OC的斜率都存在,求证:
为定值;
②试判断
的面积是否为定值,如果是,求出这个定值;如果不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cb291b5fe77e830ae19671170f72b89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
(2)如图,已知A,B,C为椭圆E上三个不同的点,原点O为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
①如果直线AB,OC的斜率都存在,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d70368971462ca516a9a9b03fbaa8e81.png)
②试判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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2023-04-24更新
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2卷引用:山东省青岛市即墨区2022-2023学年高三下学期教学质量检测数学试题
10 . 已知曲线
,直线
与曲线
交于
轴右侧不同的两点
.
(1)求
的取值范围;
(2)已知点
的坐标为
,试问:
的内心是否恒在一条定直线上?若是,请求出该直线方程;若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aff63bffc106e06fd2629a3d35c637f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d00a5df9d281dd4e1e45bf6a4d6fb27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fad20e2bc6576fc461419f8f138d26e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af2cc5f8cec8c498aa12c99c04e1c97d.png)
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2023-04-01更新
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6卷引用:山东省新高考联合质量测评2023届高三下学期3月联考数学试题
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