名校
解题方法
1 . 如图,椭圆
,圆
,椭圆C的左、右焦点分别为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/2/982fcc91-40c1-4c1c-b1b5-68f7cb1b5d18.png?resizew=179)
(1)过椭圆上一点P和原点O作直线l交圆O于M,N两点,若
,求
的值;
(2)过圆O上任意点R引椭圆C的两条切线,求证:两条切线相互垂直.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97cec1069311f0b48631d43524215250.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22fa27fccbb61045fb30f50c7b271082.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/2/982fcc91-40c1-4c1c-b1b5-68f7cb1b5d18.png?resizew=179)
(1)过椭圆上一点P和原点O作直线l交圆O于M,N两点,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac31e98551e5b14307482844b78c5498.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e177345aea68c21205d3ff3107eb1f03.png)
(2)过圆O上任意点R引椭圆C的两条切线,求证:两条切线相互垂直.
您最近一年使用:0次
2 . 在平面直角坐标系中,
,
,P为曲线E上一点,直线MP,NP的斜率之积为
.
(1)求曲线E的标准方程;
(2)过点
作直线l交曲线E于A,B两点,且点A位于x轴的上方,记直线MB,NA的斜率分别为
,
.
(ⅰ)证明:
为定值;
(ⅱ)过点B作BC垂直x轴交曲线E于不同于点A的点C,直线AC与x轴交于点D,求△ADF面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/843e3f8c3314d51a322c6122a13745c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c477662c046daefe58026249658b6d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c17c0535b686495dc04d15172da008f.png)
(1)求曲线E的标准方程;
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be92f0e0012a7696c78e3e00513edefd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
(ⅰ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67351fe10fcfc3f9072eec4c60bfaaa5.png)
(ⅱ)过点B作BC垂直x轴交曲线E于不同于点A的点C,直线AC与x轴交于点D,求△ADF面积的最大值.
您最近一年使用:0次
3 . 历史上第一个研究圆锥曲线的是梅纳库莫斯(公元前375年—公元前325年),大约100年后,阿波罗尼斯更详尽、系统地研究了圆锥曲线,并且他还进一步研究了这些圆锥曲线的光学性质:如图,从椭圆的一个焦点出发的光线或声波,经椭圆反射后,反射光线经过椭圆的另一个焦点,其中法线
表示与椭圆
的切线垂直且过相应切点的直线,已知椭圆
的中心在坐标原点,焦点为
,
,若由
发出的光线经椭圆两次反射后回到
经过的路程为
.对于椭圆
上除顶点外的任意一点
,椭圆在点
处的切线为
,
在
上的射影为
,其中
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/13/73a2984c-7d8e-4af5-b174-b473b43382fc.png?resizew=173)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/12/94838805-95a5-400e-bbcf-08ab39ca7f84.png?resizew=197)
(1)求椭圆
的方程;
(2)如图,过
作斜率为
的直线
与椭圆
相交于
,
两点(点
在
轴上方).点
,
是椭圆上异于
,
的两点,
,
分别平分
和
,若
外接圆的面积为
,求直线
的方程.
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5076829e649b3f3866d4a7e07a5713e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cddf5a506a90d09af01b81db43f17e93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/799af451da9b732f1a9164ddcf49ae13.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a6af050e5721febcc8103b9a0805399.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/13/73a2984c-7d8e-4af5-b174-b473b43382fc.png?resizew=173)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/12/94838805-95a5-400e-bbcf-08ab39ca7f84.png?resizew=197)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)如图,过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6bdac20e214b2cb3bd07f8d4778dcca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/183b6a0cef4256c9696a5bca31053da5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a12a030f853a383a50fd889486c9f42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2f9c3b578fc0598d5ec6c79404c6cc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274e29821dd4630a7182573b7c654f1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e19e695795d6c124955e2c6ec0b46056.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc26cbada73d210ae6101654e894ebc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
4 . 已知椭圆E:
的离心率为
,A,B是它的左、右顶点,过点
的动直线l(不与x轴重合)与E相交于M,N两点,
的最大面积为
.
(1)求椭圆E的方程;
(2)设
是直线AM与直线BN的交点.
(i)证明m为定值;
(ii)试堔究:点B是否一定在以MN为直径的圆内?证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d776a89f4fd29dccffe1040069d59ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a11cb104b04c4e6a1be700e81da279a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95bacae35b6e16a0a33c2bdc6bc07df7.png)
(1)求椭圆E的方程;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82aaa597a5aa6176863eda3fdf83e181.png)
(i)证明m为定值;
(ii)试堔究:点B是否一定在以MN为直径的圆内?证明你的结论.
您最近一年使用:0次
2023-03-26更新
|
798次组卷
|
3卷引用:湖南师范大学附属中学2023届高三三模数学试题
名校
解题方法
5 . 已知点
为中心在坐标原点的椭圆C的焦点,且椭圆过点
.
(1)求椭圆C的标准方程;
(2)过点F作直线
与椭圆C交于A,B两点,设
,若
,点
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e641014eb998f16c14ee0ec59f9d54f0.png)
(1)求椭圆C的标准方程;
(2)过点F作直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b25ec81508b65d234b67019c332fab0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d42bc65aa8383f8ee41a1115f01d06a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1400bea62e9e0bf6c924b796045b3948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d3d743bf924a625f564e62b9f5d8655.png)
您最近一年使用:0次
名校
解题方法
6 . 已知点
是椭圆
上一点,且在
轴上方,
,
分别为椭圆的左、右焦点,直线
的斜率为
,则
的面积为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54975582e2ef95954bb8dbd70373c6ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.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/0739793f234f8e86adc6177801ae7295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64945f76f91f91c368cd2053c3e4f9ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33d776753746914c2410a3946c357f35.png)
A.![]() | B.![]() | C.![]() | D.![]() |
您最近一年使用:0次
7 . 定义:一般地,当
且
时,我们把方程
表示的椭圆
称为椭圆
的相似椭圆.
为
上的动点,延长
至点
,使得
的垂直平分线与
交于点
,记点
的轨迹为曲线
,求
的方程;
(2)在条件(1)下,已知椭圆
是椭圆
的相似椭圆,
是椭圆
的左、右顶点.点
是
上异于四个顶点的任意一点,当
(
为曲线
的离心率)时,设直线
与椭圆
交于点
,直线
与椭圆
交于点
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be362dec96173f246ff747264007817.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/472393b18c7880e73b40e31fbe2d951c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee1845bf48cdfce200e466bd327c78e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7a18d46cc66a208677f6a3f1e79e561.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a8d03478b85373a52ed3f3706ae7b76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c3a9b723303acf1669d4d88a7172b99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d8ad9e94d07405a6be585f81a0d623b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f55620fce789bf9cd4a72ed5ba746c7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfbdbe9a17a23c44cec8c7475c4dc1a9.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)在条件(1)下,已知椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7a18d46cc66a208677f6a3f1e79e561.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b6239db84396d617f152a65098c8faf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7a18d46cc66a208677f6a3f1e79e561.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7a18d46cc66a208677f6a3f1e79e561.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a9f26eea0ada26e96bce687c9447e4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b5610f2c5b55ec06ba09cd553659794.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/5ddaa44e947bdcebc64a5f45809953de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e1e4115d78e625e9e0f47cdade3286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a45d4785a48c4e9641450e9ee2822df3.png)
您最近一年使用:0次
2023-03-03更新
|
899次组卷
|
3卷引用:湖南省长沙市雅礼中学2023届高三下学期月考(七)数学试题
8 . 已知椭圆C:
,
,
为椭圆C的左、右顶点,
,
为左、右焦点,Q为椭圆C上任意一点.
(1)求直线
和
的斜率之积;
(2)直线l交椭圆C于点M,N两点(l不过点
),直线
与直线
的斜率分别是
,
且
,直线
和直线
交于点
.
①探究直线l是否过定点,若过定点求出该点坐标,若不过定点请说明理由;
②证明:
为定值,并求出该定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cae00bdc6f8b564b6b15b32572c848b.png)
![](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/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
(1)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3853a47e9138f78e83786b0d6e85bce4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f387b16cc48e57112c89c8af2a90c1d6.png)
(2)直线l交椭圆C于点M,N两点(l不过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/448eb7d301baa90fe59b05761830f81f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a439c8a01a6626d7a3f53af31ef0bcae.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/78bb56032b37aaf40bfbac51f7fe2d9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9399c9a2a31b0e3165aea2d6ccc4f7c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b06b75fb4e379ff3b99e68f40136cad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
①探究直线l是否过定点,若过定点求出该点坐标,若不过定点请说明理由;
②证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
您最近一年使用:0次
2023-02-15更新
|
800次组卷
|
4卷引用:湖南师范大学附属中学2022-2023学年高二上学期期末数学试题
名校
解题方法
9 . 设F,E分别是椭圆
的左,右焦点,椭圆上存在点N,满足
且
的面积为20.
(1)求b的值;
(2)设点P的坐标为
,直线过点P,与椭圆交于点A,B,线段
的中点记为M.若
是
与
的等比中项,求a的最小值,并求出此时直线l的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6d71528844499d541fb06d16147a1cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a35a55ff4c434ce34d90382cea5afb85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08fffabcb0a245d00182082c8cc99349.png)
(1)求b的值;
(2)设点P的坐标为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c832f2474efe89961ef41e884da7660c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cb79ed88746f9163c7c6b0b674556de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4522fdddbca40f55f3130c4c44e0bb62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03e944586ad3d14e01f4d1ad9d8f4910.png)
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解题方法
10 . 如图,已知椭圆
与等轴双曲线
共顶点
,过椭圆
上一点P(2,-1)作两直线与椭圆
相交于相异的两点A,B,直线PA、PB的倾斜角互补,直线AB与x,y轴正半轴相交,分别记交点为M,N.
(2)若直线AB与双曲线
的左,右两支分别交于Q,R,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21e6314e24c0225d455415c52124052b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e04c8d7fbd6165d240cad25eaef7b8ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)若直线AB与双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fbac86947f24d80503333eed69e0427.png)
您最近一年使用:0次
2023-01-02更新
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1330次组卷
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2卷引用:湖南省长沙市雅礼中学2022-2023学年高三上学期月考(五)数学试题