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1 . 已知曲线C的方程为
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88895a2663c11f82b2de3abcfb1ba4c3.png)
A.存在实数![]() ![]() |
B.若曲线C为椭圆,则![]() |
C.若曲线C为焦点在x轴上的双曲线,则![]() |
D.当曲线C是椭圆时,曲线C的焦距为定值 |
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2023-12-09更新
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5卷引用:安徽省淮南第一中学2023-2024学年高二下学期开学考试数学试题
解题方法
2 . 过椭圆
的焦点
,且垂直于长轴的弦长为
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ef68992bd2a57c6dd93ad36c6a25497.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ce1a7cec7d918ce03c607f4ad8b50fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://img.xkw.com/dksih/QBM/2023/3/10/3191772901810176/3200952318615552/STEM/8445943524f542cd96e4fb221678fffc.png?resizew=4)
A.椭圆方程为![]() |
B.椭圆方程![]() |
C.过焦点![]() ![]() ![]() |
D.过焦点且长度为![]() |
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解题方法
3 . 已知椭圆
:
上的点
到左、右焦点
,
的距离之和为4.
(1)求椭圆
的方程.
(2)若在椭圆
上存在两点
,
,使得直线
与
均与圆![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2beaf83ba7b508fbfa424aa08dfaca86.png)
相切,问:直线
的斜率是否为定值?若是定值,请求出该定值;若不是定值,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e90064d012356de1877aa697cd6d6ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若在椭圆
![](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/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d454c82d9e52747563d47b68099249.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2beaf83ba7b508fbfa424aa08dfaca86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/445489a1d1c2f9a53608c7121d42f02b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
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2023-03-05更新
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4 . 吹奏乐器“埙”(如图1)在古代通常是用陶土烧制的,一种埙的外轮廓的上部是半椭圆,下部是半圆.半椭圆
(
,
且为常数)和半圆
组成的曲线
如图2所示,曲线
交
轴的负半轴于点
,交
轴的正半轴于点
,点
是半圆上任意一点,当点
的坐标为
时,
的面积最大,则半椭圆的方程是()
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/6/7f2bd8b3-1534-4054-b431-bc123e63a617.png?resizew=264)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc35409c8054fe18431c70e6fea0334.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc65d6eb9b63f96d80b54ec9893aee8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0961f5c2f41404dd1fd52af54d8e9641.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa664d429ffb252f4ea82d369fb06621.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b1d9c8a686d42030125a6209e5f37dd.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/6/7f2bd8b3-1534-4054-b431-bc123e63a617.png?resizew=264)
A.![]() | B.![]() |
C.![]() | D.![]() |
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解题方法
5 . 已知椭圆
的上顶点为A,左、右焦点分别为
、
,三角形
的周长为6,面积为
.
(1)求椭圆C的方程;
(2)已知点M是椭圆C外一点,过点M所作椭圆的两条切线互相垂直,求三角形
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.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/0530e3288e75edc196427ebc1448f201.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
(1)求椭圆C的方程;
(2)已知点M是椭圆C外一点,过点M所作椭圆的两条切线互相垂直,求三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b3ed1eca414c4ce28f85d1b0fd3a1e6.png)
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解题方法
6 . 已知M,N两点的坐标分别为
,直线MQ,NQ相交于点Q,且它们的斜率之积为
.
(1)求点Q的轨迹方程;
(2)设过点
的直线l与点Q的轨迹交于D,E两点,问在x轴上是否存在定点P,使得
为常数?若存在,求出点P的坐标以及该常数的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0586ffe4bc516265086c6b5eafd1eed7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(1)求点Q的轨迹方程;
(2)设过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f55d12701014cf53071093e8739d089b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae88a1970c0af500b31bf366e30fff98.png)
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7 . 已知椭圆C:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
的离心率为
,且过点
.
(1)求C的标准方程;
(2)过C的左焦点且斜率为
的直线l与C交于A,B两点,O为坐标原点,当
的面积为
时,求k的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/434249d6640b0c1a712d215cf8b83d5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2438dab7a692bb89a637fc9932a5c476.png)
(1)求C的标准方程;
(2)过C的左焦点且斜率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2799abb64fd7bfce9dfa7228aa460564.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/292b791c7cf21c172e6e7f97f04be176.png)
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2023-02-15更新
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解题方法
8 . 已知椭圆
的离心率为
,且点
在椭圆上.
(1)求椭圆的方程;
(2)过椭圆右焦点
作两条互相垂直的弦AB与CD,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e67baac84cf5c95d06d50c36cab7c68.png)
(1)求椭圆的方程;
(2)过椭圆右焦点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/532fb9d8a0715b513e9e3f144ea264e9.png)
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2023-02-06更新
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5卷引用:安徽省宿州市泗县第一中学2022-2023学年高二下学期开学考试数学试题
安徽省宿州市泗县第一中学2022-2023学年高二下学期开学考试数学试题 浙江省绍兴市第一中学2023-2024学年高二上学期期末考试数学试卷湖南师范大学附属中学2022-2023学年高三上学期月考(五)数学试题(已下线)江苏省八市2023届高三二模数学试题变式题17-22(已下线)重难点突破06 弦长问题及长度和、差、商、积问题(七大题型)-1
9 . 已知点
在椭圆
上,椭圆C的左右焦点分别为
,
,
的面积为
.
(1)求椭圆C的方程;
(2)设点A,B在椭圆C上,直线PA,PB均与圆
相切,记直线PA,PB的斜率分别为
,
.
(i)证明:
;
(ii)证明:直线AB过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3c9708ef0dc6d6f5dcf6596d3e4f6e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.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/33d776753746914c2410a3946c357f35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/839c7616cd0d90265f4b2c9c021254fe.png)
(1)求椭圆C的方程;
(2)设点A,B在椭圆C上,直线PA,PB均与圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b24a6cf2939489ab71d151b5a70faa4.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/46b728c0e69820cdcd839e67ffdb1014.png)
(ii)证明:直线AB过定点.
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9卷引用:安徽省舒城中学2023-2024学年高二下学期开学考试数学试卷
安徽省舒城中学2023-2024学年高二下学期开学考试数学试卷(已下线)3.1.2 椭圆的几何性质(难点)-2022-2023学年高二数学《基础·重点·难点 》全面题型高分突破(苏教版2019选择性必修第一册)第二章 平面解析几何章末检测(能力篇)(已下线)第3章 圆锥曲线与方程(A卷·知识通关练)(3)黑龙江省伊春市伊美区第二中学2022-2023学年高二上学期期中考试数学试题(已下线)第25讲 圆锥曲线直线圆过定点问题-【同步题型讲义】2022-2023学年高二数学同步教学题型讲义(人教A版2019选择性必修第一册)山东省青岛市2022届高三下学期5月二模考试数学试题(已下线)专题22 圆锥曲线中的定点、定值、定直线问题 微点4 圆锥曲线中的定点、定值、定直线综合训练(已下线)专题9-5 圆锥曲线大题基础:定点归类
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解题方法
10 . 已知椭圆
的左右焦点分别为
,
,点
是椭圆上一点,
是
和
的等差中项.
(1)求椭圆的标准方程;
(2)若A为椭圆的右顶点,直线AP与y轴交于点H,过点H的另一直线与椭圆交于M、N两点,且
,求直线MN的方程.
![](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/a216c108cb54c2b3845ac7dce1ed10a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cdfc6a59392d1ac3cd89ddc0308864c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4ef569668e797b1e94257fd5f4384dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96fa27f8db6167d4802a510371077bb5.png)
(1)求椭圆的标准方程;
(2)若A为椭圆的右顶点,直线AP与y轴交于点H,过点H的另一直线与椭圆交于M、N两点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cecff51d30a0a4b71328d7a062e83b0a.png)
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