名校
解题方法
1 . 已知抛物线
,过点
作不与x轴垂直的直线
,
分别与抛物线C交于M,N和P、Q两点.
(1)若M,N两点的纵坐标之和为-6,求直线l的斜率;
(2)证明:
;
(3)若点E为线段MN的中点,点G为线段PQ的中点,求
的值.
注:k表示直线的斜率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb4dd4670828f75bc573b52cdd02e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e53efcba08b1c0c9f7b495c2a289ebea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
(1)若M,N两点的纵坐标之和为-6,求直线l的斜率;
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90c719342ee48eb3c22ae4c78f5e325e.png)
(3)若点E为线段MN的中点,点G为线段PQ的中点,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0465c3a283ffcf2b26097c9a8e3d380.png)
注:k表示直线的斜率.
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2卷引用:广东省部分地市2023届高三下学期模拟(三)数学试题
2 . 如图,已知
是抛物线
的焦点,过点
和点
分别作两条斜率互为相反数的直线
,交抛物线于
四点,且线段
相交于点
,则下列选项中正确的是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/15/a90c4510-de6d-44c4-836c-67c6c9fb292d.png?resizew=149)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb4dd4670828f75bc573b52cdd02e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea5b6926cdb99eb58070b8720718f71a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c4cd264c97c1f261229925cc5a6761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/15/a90c4510-de6d-44c4-836c-67c6c9fb292d.png?resizew=149)
A.![]() | B.![]() |
C.![]() | D.![]() |
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名校
解题方法
3 . 抛物线:
,
是
上的点,直线
与
交于
两点,过
的焦点
作
的垂线,垂足为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d087313c8bc308b2a5832d6d3dd85174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d93594c33ce81ec8b3b958ce9500ebf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
A.![]() | B.![]() |
C.![]() | D.若![]() ![]() ![]() ![]() |
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3卷引用:福建省厦门市2023届高三毕业班第四次质量检测数学试题
4 . 在平面直角坐标系中,P,Q是抛物线
上两点(异于点O),过点P且与C相切的直线l交x轴于点M,且直线
与l的斜率乘积为
.
(1)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
(2)过M作l的垂线交椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c82e7d9f4f7ace849e09e9adcb786b7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea1f0417d8269f01d8e0bc1a8756e2ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
①当取最大值时,求点P的纵坐标;
②证明:存在定点G,使为定值.
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5卷引用:山东省烟台市2023届高考适应性练习(一)数学试题
山东省烟台市2023届高考适应性练习(一)数学试题山东省枣庄市2023届高三三模数学试题湖南省株洲市第二中学2022届高三下学期第三次月考数学试题(已下线)高二上学期期中复习【第三章 圆锥曲线的方程】十二大题型归纳(拔尖篇)-2023-2024学年高二数学举一反三系列(人教A版2019选择性必修第一册)(已下线)通关练17 抛物线8考点精练(3)
5 . 已知抛物线
与双曲线
相交于两点
是
的右焦点,直线
分别交
于
(不同于
点),直线
分别交
轴于
两点.
(1)设
,求证:
是定值;
(2)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b519a71ba66154cd5c38a06977d70e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bc6ea0beb6741bf242f67a6eedf9baa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/016e82617e4586c46e55b27cd604db1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21e9feabc99f62ee569b460e61526e2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2013303974d530fce9a9930938f4c2b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28199a0d8596dd0ad472bd807346c81d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a85af6eaff9a4e3e9af4e9c1f4f7b996.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a15732502fe6599bcfe364f0d6d841b.png)
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4卷引用:浙江省温州市2023届高三下学期5月第三次适应性考试(三模)数学试题
6 . 已知点
在抛物线
上,过点
的直线
与
相交于
两点,直线
分别与
轴相交于点
.
(1)当弦
的中点横坐标为3时,求
的一般方程;
(2)设
为原点,若
,求证:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a865d0cdbd1f950ba14112528db7af2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fef26f14e6a51c897b4dd93c72ad133.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/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03df57efff473b3cfeb8503796b7d6b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e1e4115d78e625e9e0f47cdade3286.png)
(1)当弦
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9b19c26574ddeb2154b2cbc2f755a8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bed3650dd045ee37623cefbea36f17c9.png)
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7 . 已知A,B是抛物线E:
上不同的两点,点P在x轴下方,PA与抛物线E交于点C,PB与抛物线E交于点D,且满足
,其中λ是常数,且
.
(1)设AB,CD的中点分别为点M,N,证明:MN垂直于x轴;
(2)若点P为半圆
上的动点,且
,求四边形ABDC面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45952ab761aa81ae497dbca35aa80e96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/472393b18c7880e73b40e31fbe2d951c.png)
(1)设AB,CD的中点分别为点M,N,证明:MN垂直于x轴;
(2)若点P为半圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07f35cedbcc67171250cde00f5c7f7c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f2f2d7c81cb44416bcdf59419637682.png)
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2023-04-27更新
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6卷引用:广东省2023届高三二模数学试题
广东省2023届高三二模数学试题(已下线)专题06 解析几何专题20平面解析几何(解答题)(已下线)模块六 专题7易错题目重组卷(广东卷)河南省信阳市浉河区信阳高级中学2024届高三上学期第八次大考数学试题(已下线)黄金卷02(2024新题型)
解题方法
8 . 已知抛物线C:
的焦点为F,点
,且
.
(1)求抛物线C的方程;
(2)过点Q作直线l交C于A,B两点,O为原点,过点A作x轴的垂线,分别与直线
,
交于点D,E,从下面①②两个问题中选择一个作答.
①问:
是否为定值,并说明理由;
②问:在直线
上是否存在点M,使四边形
为平行四边形,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f21443bffcdb46d7762a351c4d09bb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf8d1128232dffdd5fbe3f9d81d50aee.png)
(1)求抛物线C的方程;
(2)过点Q作直线l交C于A,B两点,O为原点,过点A作x轴的垂线,分别与直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
①问:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52b1851b0c87ffde837e3540a3c5d680.png)
②问:在直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5489dcbeab40a8d1c0f5af0a6219e3.png)
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9 . 已知抛物线
的焦点为F,过抛物线上任意一点P作圆
的切线
,A为切点,且直线
交抛物线于另一点Q,则下列结论正确的有( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/26/3eb91c44-acf4-4b00-ba15-1f5d4646e9bf.png?resizew=195)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e8953ded144195804384dcb494d5e2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f11583d5a9d06b238a94e3f24c8aa41b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/26/3eb91c44-acf4-4b00-ba15-1f5d4646e9bf.png?resizew=195)
A.![]() ![]() |
B.![]() ![]() |
C.三角形![]() ![]() |
D.连接![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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3卷引用:微点5 塞瓦定理、富瑞基尔定理、笛沙格定理、彭塞列闭合定理综合训练
(已下线)微点5 塞瓦定理、富瑞基尔定理、笛沙格定理、彭塞列闭合定理综合训练山东省菏泽市2022-2023学年高二下学期2月教学质量检测数学试题安徽省合肥市第一中学2023-2024学年高二上学期1月考数学考试试题
10 . 已知O为坐标原点,F为抛物线
的焦点,C的准线与x轴的交点为
,过F的直线l与C交于A,B两点,与C的准线交于点E,直线l的倾斜角
,且点A在第一象限,下列选项正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7ff329f3b12cf5678e99941e7188621.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/480888079e87df2b1c0734114a594d3e.png)
A.![]() | B.![]() |
C.若F为AE的中点,则![]() | D.若B为AE的中点,则![]() |
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2023-04-21更新
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2卷引用:河北省承德市2023届高三下学期4月高考模拟数学试题