1 . 已知椭圆:
:
的离心率
,连接椭圆的四个顶点得到的菱形的面积为
.
,
是椭圆
的两个焦点.
(1)求椭圆
的方程;
(2)设
为E的左顶点,过
点作两条互相垂直的直线
分别与E交于
,
两点,证明:直线
经过定点,并求这个定点的坐标.
(3)设
是椭圆
上一点,直线
与椭圆
交于另一点
,点
满足:
轴且
,求证:
是定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5de85df85401e7e8da683ea4a784963c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.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/2a30f3a8b673cc28bd90c50cf1a35281.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97bde5efa645a4c1ed6874088400d6a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
(3)设
![](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/9ac86e1c253297a377e14fb9a1689be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2784a52c4da98dc9df661fc152fc29e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23d6de923fac9e8a6bccfb8e2b68a4bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b57abf0dc1e0bf1a11dbef810607de18.png)
您最近一年使用:0次
名校
解题方法
2 . 已知椭圆
的离心率为
,设
是C上的动点,以M为圆心作一个半径
的圆,过原点作该圆的两切线分别与椭圆C交于点P、Q,若存在圆M与两坐标轴都相切.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/4/ae0ddc2f-2c19-4cc2-a1fa-ceb098ce66ef.png?resizew=175)
(1)求椭圆C的方程;
(2)若直线OP,OQ的斜率都存在且分别为
,
,求证:
为定值;
(3)证明:
为定值?并求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22962a2ad892cb6b14ab039a06e8cdc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8176754726d2194c890e80df1a1f1c3a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/4/ae0ddc2f-2c19-4cc2-a1fa-ceb098ce66ef.png?resizew=175)
(1)求椭圆C的方程;
(2)若直线OP,OQ的斜率都存在且分别为
![](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/b4757181824e15e0f21e5bdd55448783.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5f6c5fd93aed88bec58002a20ea2e90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bab77b1212086d7b16e288f73a09560.png)
您最近一年使用:0次
2022-12-03更新
|
594次组卷
|
3卷引用:四川省成都市树德中学2022-2023学年高二上学期期中考试数学(文)试题
名校
解题方法
3 . 已知椭圆
过点
,且离心率为
.设
,
为椭圆
的左、右顶点,
为椭圆上异于
,
的一点,直线
,
分别与直线
相交于
,
两点,且直线
与椭圆
交于另一点
.
(1)求椭圆
的标准方程;
(2)求证:直线
与
的斜率之积为定值;
(3)判断三点
,
,
是否共线?并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.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/c5db41a1f31d6baee7c69990811edb9f.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/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/495bb3e5a3a9d35f5c9f0cf1f5d51876.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/8e5c62f22d7afc5627fcb86599faa8e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
(3)判断三点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
您最近一年使用:0次
4 . 已知椭圆
的左右顶点
的坐标分别为
且椭圆E的离心率为
.
(1)求椭圆E的方程;
(2)过点
作直线l交椭圆E于P,Q两点,且点P位于x轴上方,记直线
的斜率分别为
.
①证明:
;
②设点Q关于x轴的对称点为
,求证直线
过x轴上一个定点,并求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e5578ca83f5bd5c285994061b9c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00442d96d695db2c58bf1fb7165fca94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0586ffe4bc516265086c6b5eafd1eed7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(1)求椭圆E的方程;
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2ba2238d6afe0187534155dd9ac48c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e990f8a065b517a4353903abe94ef4df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c80b04ce48c9ace43276552c77108126.png)
②设点Q关于x轴的对称点为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a86380a6d6501f6504dcb4aa5e3099f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df356846992174e641898c61380af591.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c2cf4b11fe84c33c931919391e705c5.png)
您最近一年使用:0次
名校
解题方法
5 . 已知椭圆
的左、右焦点分别为
,
,离心率为
,过椭圆的左焦点
作不与x轴重合的直线MN与椭圆相交于M,N两点,
的周长为8,过点M作直线
的垂线ME,E为垂足.
(1)求椭圆C的标准方程;
(2)证明:直线EN经过定点P,并求定点P的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.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/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5585a42c8f07ad90b94ace9db3d78994.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae65a3b431bc5e09408beb3466908d6a.png)
(1)求椭圆C的标准方程;
(2)证明:直线EN经过定点P,并求定点P的坐标.
您最近一年使用:0次
名校
6 . 如图,由部分椭圆
和部分双曲线
,组成的曲线
称为“盆开线”.曲线
与
轴有
两个交点,且椭圆与双曲线的离心率之积为
.
的直线
与
相切于点
,求部分椭圆方程、部分双曲线方程及直线
的方程;
(2)过
的直线
与
相交于点
三点,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/573d0b25ac3ea513b454e803dc9b67a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1248f0dcc3786ba984dc7fc88cf62e8f.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/526e16c3287f8195d925b3805e0b3c1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/059d02ae074c7c2f7dfde8058dfa55ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff32d26c8d44f5fb4813a19c1030a9f5.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/762c7b2136f8428ff7ec20fd8efd03c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.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/4bf70a4a36f163ceee458c6963b2bee5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29fda29f1b1b042a89c0213f18d341ad.png)
您最近一年使用:0次
解题方法
7 . 在平面直角坐标系
中,椭圆
的右顶点和上顶点分别为
,点
是直线
上的动点,设直线
斜率分别为
.
(1)求椭圆
的离心率;
(2)求证:
为定值;
(3)若直线
与椭圆的另一个交点分别为
,试判断直线
与直线
的位置关系.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7795aec93c2c7ac2fd93e6747ca6516c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb4402aeb853b22f20992156957ef0fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b87609b100b8d39b52e25ef1bee9b772.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/790ef3382b1c731f2885eecfd92c2a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4757181824e15e0f21e5bdd55448783.png)
(3)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/790ef3382b1c731f2885eecfd92c2a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
您最近一年使用:0次
8 . 已知椭圆
:
,
,
为左右焦点,直线l过左焦点
与椭圆交于A,B两点,其中A在第一象限,记
,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/26/2407d664-f5bd-480f-b43b-0084f5d2156c.png?resizew=173)
(1)若椭圆
的离心率为
,三角形
的周长为6,求椭圆
的方程;
(2)求证:
;
(3)直线
与椭圆交于另一点
,若
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.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/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57654bf131df03b31d3d11b5b656c73d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb57bb18755127be041d346444a4743e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe197f95a36f68ee80f69ff5f4a26970.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/26/2407d664-f5bd-480f-b43b-0084f5d2156c.png?resizew=173)
(1)若椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/227fb6f5f485e12ddad01ef7cb1e06d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78a665b04ba8406d1f55bebb0cb7389c.png)
(3)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72404eb522c6cc692757b0855c2df865.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708d0e76f524d0e8a48db01392faac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71eda28755639d00f9d24b95679d9496.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0de82aff5049dce8a3d56291dc2ae4af.png)
您最近一年使用:0次
9 . 已知椭圆E:
的长轴为双曲线
的实轴,且离心率为
.
(1)求椭圆E的标准方程;
(2)已知椭圆
在其上一点
处的切线方程为
.过直线
上任意一点P作椭圆E的两条切线,切点分别为A,B.M为椭圆的左顶点.
①证明:直线
过定点;
②求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/347b68f42934c74e0d759a67613a1da9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dad0c75ce33673ec4c425896e8619e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(1)求椭圆E的标准方程;
(2)已知椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/347b68f42934c74e0d759a67613a1da9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2c17005d48901a9f9d996e1c7c67eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a46c2737bf9c790cdb4b767217719452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.png)
①证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a5e0a51c9e14fb246b0ba0b231c1e3.png)
您最近一年使用:0次
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解题方法
10 . 如图,已知椭圆
,长轴长为6,离心率为
,过椭圆右焦点
作斜率不为0的直线交椭圆于
、
,过
作
垂直于直线
,连接
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/14/1285412a-c035-447c-87e4-817d66631e39.png?resizew=186)
(1)求椭圆的标准方程;
(2)证明:直线
过定点,并求出定点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5adb5eb60ae4435a12d93854066298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090c0ba4cadbd85bf1f04f0d962eb16a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b15febfda66e733f14aa7115ed4343a8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/14/1285412a-c035-447c-87e4-817d66631e39.png?resizew=186)
(1)求椭圆的标准方程;
(2)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf3d566704b44ea4ef1f99c37bd46902.png)
您最近一年使用:0次
2024-01-11更新
|
518次组卷
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2卷引用:湖北省武汉市东湖中学2023-2024学年高二上学期期中考试数学试题