名校
解题方法
1 . 如图,已知
为二次函数
的图像上异于顶点的两个点,曲线
在点
处的切线相交于点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/14/b1c9ee20-7af4-4018-a49a-6703b2da8013.png?resizew=203)
(1)利用抛物线的定义证明:曲线
上的每一个点都在一条抛物线上,并指出这条抛物线的焦点坐标和准线方程;
(2)求证:
成等差数列,
成等比数列;
(3)设抛物线
焦点为
,过
作
垂直准线
,垂足为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/031da5d48fbe63745429b1add253344f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c60b6eee6448a408616e1b61bd793f0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf210c8c9e83e70f2d3ede1e18a5f3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/031da5d48fbe63745429b1add253344f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/14/b1c9ee20-7af4-4018-a49a-6703b2da8013.png?resizew=203)
(1)利用抛物线的定义证明:曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf210c8c9e83e70f2d3ede1e18a5f3d.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/297426b8f7938c8d14f42a481a19c3a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a2b440f7aac4b432fef8f4c9f8e3f76.png)
(3)设抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf210c8c9e83e70f2d3ede1e18a5f3d.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/35d58f9019097bd05037aefd5c322916.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31b7a8d232e9a11f5d471f47a1294cd4.png)
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名校
2 . 汽车前灯反射镜曲面设计为抛物曲面(即由抛物绕其轴线旋转一周而成的曲面).其设计的光学原理是:由放置在焦点处的点光源发射的光线经抛物镜面反射,光线均沿与轴线平行方向路径反射,而抛物镜曲面的每个反射点的反射镜面就是曲面(线)在该点处的切面(线).定义:经光滑曲线上一点,且与曲线在该点处切线垂直的直线称为曲线在该点处的法线.设计一款汽车前灯,已知灯口直径为20cm,灯深25cm(如图1).设抛物镜面的一个轴截面为抛物线C,以该抛物线顶点为原点,以其对称轴为x轴建立平面直角坐标系(如图2)抛物线上点P到焦点距离为5cm,且在x轴上方.研究以下问题:
(2)求P点坐标.
(3)求抛物线在点P处法线方程.
(4)为证明(检验)车灯的光学原理,求证:由在抛物线焦点F处的点光源发射的光线经点P反射,反射光线所在的直线平行于抛物线对称轴.
(2)求P点坐标.
(3)求抛物线在点P处法线方程.
(4)为证明(检验)车灯的光学原理,求证:由在抛物线焦点F处的点光源发射的光线经点P反射,反射光线所在的直线平行于抛物线对称轴.
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3 . 已知抛物线![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e97afdeaa1d4433cffe5005446fcbbbb.png)
,过焦点F的直线l与抛物线交于S,T,且
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/23/11c4f48f-a06f-431f-8779-6966df187d6e.png?resizew=187)
(1)求抛物线C的方程;
(2)设点P是x轴下方(不含x轴)一点,抛物线C上存在不同的两点A,B满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/203f6ea7bd3319d450fd957137942573.png)
,其中
为常数,且两点D,E均在C上,弦AB的中点为M.
①若点P坐标为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb3d32f46af3841f098c82e30ec1462.png)
,抛物线过点A,B的切线的交点为N,证明:点N在直线MP上;
②若直线PM交抛物线于点Q,求证;
为定值(定值用
表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e97afdeaa1d4433cffe5005446fcbbbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b75ed10db4b9747a4b6d865b774f6b6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3405289647ead6ef2e86bc2e29a29b2d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/23/11c4f48f-a06f-431f-8779-6966df187d6e.png?resizew=187)
(1)求抛物线C的方程;
(2)设点P是x轴下方(不含x轴)一点,抛物线C上存在不同的两点A,B满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/203f6ea7bd3319d450fd957137942573.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c306bcfd9225ab3637e3f307161e8f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
①若点P坐标为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb3d32f46af3841f098c82e30ec1462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdfec4233214c3a729c843dee0d186db.png)
②若直线PM交抛物线于点Q,求证;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/184d0e916ff15d13036d0c40905ab22d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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2020-01-31更新
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222次组卷
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4卷引用:【新东方】高中数学20210304-003
(已下线)【新东方】高中数学20210304-003重庆市沙坪坝区南开中学校2019-2020学年高二上学期期末数学试题(已下线)【新东方】高中数学20210323-002【高二上】重庆市南开中学2019-2020学年高二上学期期末数学试题
名校
4 . 已知实数x,y满足方程
.
(1)求
的值;
(2)设
与
是方程组
两组不同的解,其中
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1beb6812158ca2a3082bd13ca07578f0.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c1afbc87ccffbc98b9ab58df8c69bee.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99307ab4373fbe72422ae5aa980db61c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41039d45e37899d233232de3d802b105.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccee8eb181dc117834582bc433eca559.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aab3cf6695638d5bcd26580174d7cbf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1da3ff6f17be99ec311610efa08ba002.png)
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5 . 已知抛物线
经过点
中的两个点,
为坐标原点,
为焦点.
(1)求抛物线
的方程;
(2)过
且倾斜角为
的直线交
于
两点,
在第一象限,求
的值;
(3)过点
的直线
与抛物线
交于
两点,直线
分别交直线
于
两点,记直线
的斜率分别为
,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e928b6c767a9fb8f41d88ec5e81a1bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c9ea663a2ddd6ef0e00f272f84f7f72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f09757d013574cf058d5bb944fdf034a.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f09757d013574cf058d5bb944fdf034a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbbbc9c5353894f2c93c205c3ac04f03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92159c0bb7a3a9fc257109a5cd850a89.png)
(3)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e9b5e076078240e0c5ad9763a9824d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f09757d013574cf058d5bb944fdf034a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e1e4115d78e625e9e0f47cdade3286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92574df7bd90ee17a4834a608e0179e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef00713e73b8357cc7900144f5505bc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ec8858389f4c3156a946ba8bf0d8a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4757181824e15e0f21e5bdd55448783.png)
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2024高三·全国·专题练习
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解题方法
6 . 已知椭圆![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a5bbb709522dba9425a6b45ee671298.png)
的长轴长为4,一个焦点
与抛物线
的焦点重合.
(1)求椭圆
的方程;
(2)若不过
的直线
交
于
两点,使得
,求证:直线
恒过一定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a5bbb709522dba9425a6b45ee671298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/347b68f42934c74e0d759a67613a1da9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60b89db9cb2586df5d4d829c116db979.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若不过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4a30de57df4e6f60bffe9ac591b24fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fc5bd66dd6d5e09ff0893a938aed56e.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/ca2da8ff157c9f318c0a5292d2ab5648.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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2024-04-26更新
|
1002次组卷
|
4卷引用:江西省宜春市宜丰中学创新部2023-2024学年高一下学期6月月考数学试题
江西省宜春市宜丰中学创新部2023-2024学年高一下学期6月月考数学试题(已下线)FHgkyldyjsx17(已下线)第23题 解析几何有“三定”,“移植思维”建奇功(优质好题一题多解)江西省宜春市宜丰中学2023-2024学年高二下学期6月月考数学试题
23-24高二上·全国·课后作业
解题方法
7 . 已知过抛物线
的焦点F的直线交抛物线于
,
两点.求证:
(1)
,
;
(2)以AB为直径的圆与抛物线的准线相切.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/373cb7ccb8a14f1b4756700280b55f20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cfba8278d7d4e13681c828abda68df1.png)
(2)以AB为直径的圆与抛物线的准线相切.
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解题方法
8 . 如图,直线
与抛物线
相交于A,B两点.
;
(2)求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/026808536f6b6d265c778e23836fbf13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b072ff6d1b83232bebd7d4709ffba4ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3825ccc273ef9a672a606432d165b866.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4dfec890cdfdda355e19463f3be813.png)
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2023-10-06更新
|
361次组卷
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5卷引用:模块四 专题2 暑期结束综合检测2(基础卷)(人教B)
(已下线)模块四 专题2 暑期结束综合检测2(基础卷)(人教B)湘教版(2019)选择性必修第一册课本习题 习题3.3山东省泰安市2022-2023学年高二上学期期末数学试题(已下线)3.3 抛物线(已下线)专题27 抛物线的简单几何性质7种常见考法归类 - 【考点通关】2023-2024学年高二数学高频考点与解题策略(人教A版2019选择性必修第一册)
9 . 已知
,
,
为椭圆
上三个不同的点,满足
,其中
.记
中点
的轨迹为
.
(1)求
的方程;
(2)若直线
交
于
,
两点,交
于
,
两点,求证:
.
![](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/0ae6de03e2b3bef75237eb998d6e11d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d4388577dc452ba9d9caf47d81e6f10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4582da4e864cc32f1ad5af71769be480.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.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/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d2c4c64fb2c72ee592aa9da98ad6b20.png)
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2023-05-27更新
|
688次组卷
|
3卷引用:江西省宜春市丰城市第九中学2023-2024学年高一上学期第三次月考数学试题
名校
解题方法
10 . 已知椭圆
的离心率为
,长轴的左端点为
.
(1)求C的方程;
(2)过椭圆C的右焦点的任一直线l与椭圆C分别相交于M,N两点,且AM,AN与直线
,分别相交于D,E两点,求证:以DE为直径的圆恒过x轴上定点,并求出定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd99c5000629d7f49499d666e68f40d.png)
(1)求C的方程;
(2)过椭圆C的右焦点的任一直线l与椭圆C分别相交于M,N两点,且AM,AN与直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.png)
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2023-04-06更新
|
1356次组卷
|
7卷引用:江西省景德镇一中2022-2023学年高一(19班)下学期期中考试数学试题.
江西省景德镇一中2022-2023学年高一(19班)下学期期中考试数学试题.北京市门头沟区2023届高三综合练习(一)数学试题专题10平面解析几何(非选择题部分)北京卷专题23平面解析几何(解答题部分)北京市第八中学2023-2024学年高三下学期零模练习数学试题(已下线)湖北省“荆、荆、襄、宜四地七校”考试联盟2023-2024学年高二下学期期中联考数学试卷变式题16-19天津市河西区2023-2024学年高三下学期总复习质量调查(三)数学试卷