名校
1 . 设椭圆
的一个焦点为
,点
为椭圆
内一点,若椭圆
上存在一点
,使得
,则椭圆
的离心率的取值范围是__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e5578ca83f5bd5c285994061b9c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2ba2238d6afe0187534155dd9ac48c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21f20953302d861e6c698575bfbab1c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294d1547a7b99c880d47ce660447b4b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
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2 . 在棱长为1的正方体ABCD − A′B′C′D′中,若点P 是棱上一点,则满足
的点P 的个数为_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f95f087b7f5ab6f8bd4a197df4ee3e0.png)
![](https://img.xkw.com/dksih/QBM/2018/1/8/1856066903818240/1858751544705024/STEM/b6a759d1d15044a496f629b7ffa5defb.png?resizew=172)
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3 . 已知椭圆
的左、右焦点分别为
、
,抛物线
的焦点为
,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1817297374e431a1d9087dcc2991d3a9.png)
________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0dce10bf671cc2ebc67aa7fd568a9a9.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/e464e56ccbc33ed31393e8bff32ac66d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcf30286863260873e9019773c7dc2a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1817297374e431a1d9087dcc2991d3a9.png)
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名校
4 . 平面内与两定点
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f43ae7ce75120f481eb5397376ba5af.png)
连线的斜率之积等于非零常数
的点的轨迹,加上
、
两点所成的曲线
可以是圆、椭圆或双曲线.给出以下四个结论:
①当
时,曲线
是一个圆;②当
时,曲线
的离心率为
;
③当
时,曲线
的渐近线方程为
;
④当
时,曲线
的焦点坐标分别为
和
.其中全部正确结论的序号为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1431b1f091ddf2abf39c3d18baffd09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f43ae7ce75120f481eb5397376ba5af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bf8197e4f3fd18815045d29c357a863.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.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/c5db41a1f31d6baee7c69990811edb9f.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aed39f5aca78934fb383402433fe549.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0db2c49919467a2e14540f2aabd05cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
③当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e94f16d5ed858699bfea5039a7bf8ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64a5d290a9fda51acf454c8fc893f9a4.png)
④当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80d0a2348c50b0f6facdc960518d3601.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9665fc8298cc23cff367a5ad921f492d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b9ab14908726a901332dfcb27f8f7a0.png)
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5 . 如图,两个椭圆
、
内部重叠区域的边界记为曲线
是曲线
上的任意一点,给出下列四个判断:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/23/b7f07cff-39b1-4e64-875d-9e43821bfa3d.png?resizew=157)
①
到
、
、
、
四点的距离之和为定值;
②曲线
关于直线
均对称;
③曲线
所围区域面积必小于36.
④曲线
总长度不大于6π.上述判断中正确命题的序号为________________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b747db7eaf469c6d1607e4b0d028299f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/781bf1ba5437d9095fe95474f1260376.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99e11127fbf96867f29f22cda3e775a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/23/b7f07cff-39b1-4e64-875d-9e43821bfa3d.png?resizew=157)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a9659fe1279984da6a8d4d2bc369142.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/682f50684ae25e45dfec0eaf8112a7a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5afc8f55e1265355310db729d5b8026c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4efb6f8e0a4b33da45f2cc157c7ec015.png)
②曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2816c1cb72ed019c7a062bd59aab4af9.png)
③曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
④曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
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6 . 已知
是椭圆和双曲线的公共焦点,
是它们一个公共点,且
,椭圆、双曲线的离心率分别为
,则
的最小值__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db634c178cd7bffbd4cb886e3f2cca22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f994a876391534efe497dc115a53e3fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb70c73df1696ee91cfc72e1a57f6dec.png)
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7 . 已知两个等高的几何体在所有等高处的水平截面的面积相等,则这两个几何体的体积相等. 椭球体是椭圆绕其轴旋转所成的旋转体. 如下图将底面直径皆为
,高皆为
的椭半球体及已被挖去了圆锥体的圆柱体放置于同一平面
上. 以平行于平面
的平面于距平面
任意高
处可横截得到
及
两截面,可以证明
总成立. 则短轴长为
,长轴为
的椭球体的体积为__________
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18436f0e2391b0ab7537a566fc28204c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1412bb5c926c15b192eefe0795015074.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd79498dbcdfc8f158ac6acd69cdb133.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81a5521fd7492c1a325a423571dee25c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/095ab4a92bf822e175d370e6d0c8a730.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dd6f4250ca6b1b9bce234a01f00d44d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc6d1d99afa158b4ba4fc0dae562fcc1.png)
![](https://img.xkw.com/dksih/QBM/2018/1/8/1856051970883584/1856741347090432/STEM/69fc502dcd0d4e8ca11d284cf4c35a0b.png?resizew=288)
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名校
解题方法
8 . 已知
为椭圆
的右焦点,点
,点
为椭圆上任意一点, 且
的最小值为
,则
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4547d22964b5851ec3b7d270d9e25e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32366143230ca122894a4bada7c7b96d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdeb51d4c591643b189d102d31648874.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/354c3a283b2b21cc8ac33995aac20a5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c57bbef89a37f1a3808c0ceeac0c22.png)
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9 . 已知
为椭圆
的两个焦点,点
在椭圆上,且
的面积为
,则
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfe5b92288eb440732047addaa59e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ec56b59d6f2654570c2b5c4fd13a05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ec92060d3d3248137cb49a10db7ed69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2de987724d4f57a09be3d6edd652443.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a2dcb8faf9a7df7e884850a4e37b4e9.png)
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解题方法
10 . 过点
的直线
交椭圆
于
两点,
为椭圆的左焦点,当
周长最大时,直线
的方程为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b25e326fdf9e5456f48e8a99a069f379.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ef0245b050242408160f00ce19f92c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/635a0150fe965ced8935d7479b22c508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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