1 . 已知抛物线
:
及抛物线
:
(
),过
的焦点F的直线与
交于
,
两点,与
交于
,
两点,O为坐标原点,
.
(1)求
的方程.
(2)过
的中点M作
的准线的垂线,垂足为N.
(ⅰ)证明:
为定值;
(ⅱ)判断直线
与
的公共点个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fc58c62444bf42a25289c45425a00f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5abd313d4e92a762fb7fb0c1cb65263d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a71fc9c0068109dad1382354570665.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7fd372b6c8b652570d9307d98c6b4e1.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc34876d748f30fa4fc2eb6a686b5ff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(ⅰ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27a7397aff70bff06b1dc694cf67482e.png)
(ⅱ)判断直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b06b75fb4e379ff3b99e68f40136cad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
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2 . 在直角坐标系
中,点
到直线
的距离等于点
到点
的距离,记动点
的轨迹为
.
(1)求
的方程;
(2)直线
与
交于
两点,求线段
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99c6875d552e9fff3c7d655f3a59b166.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe1847aa6c2efd56c8f2bf7a6effe523.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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3卷引用:海南省海南高二期末考试2023-2024学年高二上学期1月期末数学试题
海南省海南高二期末考试2023-2024学年高二上学期1月期末数学试题(已下线)2.4.1 抛物线的标准方程(十四大题型)(分层练习)-2023-2024学年高二数学同步精品课堂(沪教版2020选择性必修第一册)河北省石家庄精英中学2023-2024学年高二上学期期末数学试题
解题方法
3 . 已知椭圆
与直线
相交于两个不同的点
,点
为线段
的中点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb4402aeb853b22f20992156957ef0fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1a8ba78658831f9fce13aafa1d93973.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
A.![]() | B.![]() ![]() |
C.弦长![]() ![]() | D.点![]() ![]() |
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4 . 已知点
在抛物线
的准线上,过点
作直线
与抛物线
交于
两点,斜率为2的直线与抛物线交于
两点.
(1)求抛物线
的标准方程;
(2)① 求证:直线
过定点
;
② 若
的面积为
,且满足
,求直线
斜率的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbc2de9df8bce6139613bb86322db0f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.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/e52586ca2a3b783bc8092415e2d4bf6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e478787ebfeb68a5a7594dbd9eecd4.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)① 求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
② 若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f7ad41b36674fd6e90176ee24cdefbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41c8bef41fa6778e5eb57e2f19ea48f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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2024-01-27更新
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3卷引用:湖北省武汉外国语学校2023-2024学年高二上学期期末考试数学试题
湖北省武汉外国语学校2023-2024学年高二上学期期末考试数学试题(已下线)2.4.2 抛物线的性质(九大题型)(分层练习)-2023-2024学年高二数学同步精品课堂(沪教版2020选择性必修第一册)江苏省徐州市沛县第二中学2024届高三下学期期初测试数学试题
5 .
表示以点
为中心的椭圆,如图所示,
为椭圆C:
的左焦点,Q为直线
上的一点,P为椭圆C上的一点,以
为边作正方形
(F,P,A,B按逆时针排列),当P在椭圆上运动时,
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc5d275721db46e39e5b23a7f2b64268.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9799b1b1f49ef15916b0018547564590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caa04eb7cae40ce5e89da590b9cdb75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/134dde7006742eba3252cb11fe959abe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98f739b07f992bce672aef2fc48b4b93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c2293f93791a597bf0162411f3395f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/857e804aac2f81757178d237485299fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b65ea04098e8d2809d8163f27d49860c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/17/cf3db5f3-5935-4e04-8948-1081f9a525e9.png?resizew=187)
您最近一年使用:0次
解题方法
6 . 已知双曲线
的左、右顶点分别为
,离心率为
.过点
的直线l与C的右支交于M,N两点,设直线
的斜率分别为
.
(1)若
,求
;
(2)证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e9c11cc36320090d0aaf0c621a63b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31f8f7e40ba386c0a9675896b52752d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00ed24bfcc37b79fe9ca61ed8fdf26ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92e4cfa2db814e7e2e8abdf033706420.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dca1726d463bd741c904abd9b6589056.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e607c2f7ae494246768832acb0d54a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbf434334b09cc0fdd4e86e84e6ceb00.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3342ecc389773431bc3f64410e41191.png)
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解题方法
7 . 已知椭圆
,F为右焦点,A为右顶点,B为上顶点,
.
(1)求C的离心率e;
(2)已知MN为C的一条过原点的弦(M,N不同于点A).
(ⅰ)求证:直线AM,AN的斜率之积为定值,并求出该值;
(ⅱ)若直线AM,AN与y轴分别交于点D,E,且△ADE面积的最小值为
,求椭圆C的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c93e065b3576b3cd585c5fe8410ebfe4.png)
(1)求C的离心率e;
(2)已知MN为C的一条过原点的弦(M,N不同于点A).
(ⅰ)求证:直线AM,AN的斜率之积为定值,并求出该值;
(ⅱ)若直线AM,AN与y轴分别交于点D,E,且△ADE面积的最小值为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
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2024-01-25更新
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82次组卷
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2卷引用:河南省开封市2023-2024学年高二上学期期末调研考试数学试卷
8 . 已知双曲线
的左、右顶点分别为
,右焦点为
,
为
上异于顶点的动点,则下列说法正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb50abb72f53004ab44468777b29f294.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
A.双曲线![]() ![]() |
B.双曲线![]() ![]() |
C.点![]() |
D.直线![]() ![]() ![]() |
您最近一年使用:0次
名校
解题方法
9 . 已知实数
,
满足
,则
的取值范围是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e83a76009344ccdac9994749f23d4108.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02fe1b6f20cd9115202e23090f1ea9da.png)
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2卷引用:江西省吉安市2023-2024学年高二上学期期末教学质量检测数学试题
名校
解题方法
10 . 已知抛物线
,圆
,过点M的直线l与E交于A,B两点,与圆M交于C,D两点(A,C都在x轴上方),若
,则直线l的斜率为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f4fb72e39d79b7a0cd892fa5fa34bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/948811debe828137d751f874107b6cfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc34f9174c6f52d6f669e775e865a25c.png)
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2024-01-24更新
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233次组卷
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4卷引用:2.4.2 抛物线的性质(九大题型)(分层练习)-2023-2024学年高二数学同步精品课堂(沪教版2020选择性必修第一册)
(已下线)2.4.2 抛物线的性质(九大题型)(分层练习)-2023-2024学年高二数学同步精品课堂(沪教版2020选择性必修第一册)河北省唐山市2024届高三上学期期末数学试题河北省衡水市枣强县名校协作2024届高三上学期期末数学试题湖南省长沙市宁乡市第一高级中学2021届高三第三次模拟考试数学试卷