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1 . 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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解题方法
2 . 在平面直角坐标系中,已知
为坐标原点,点列
,直线系
,
,若直线
与直线
交于点
.
(1)求证:点
在抛物线上,并求出该抛物线的方程;
(2)设
,
为(1)中抛物线上两个不同的点,直线
,
的斜率分别为
,
,且
,证明:直线
经过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bebe8cf09b5ddaa37deabcb0599e1193.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/230da62e505c4dfd3dbbd38f7311abc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c7c5e7bd6bac51402ffa04b4144ec78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9123b1c8cb0bbaca44e8464bee03678.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
(1)求证:点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
(2)设
![](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/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f745dbb0d0e93c05041935ea760eb50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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2021-01-02更新
|
302次组卷
|
5卷引用:河北省易县中学2020-2021学年高二上学期12月月考数学试题
3 . 如图,已知抛物线与坐标轴分别交于A
、B
、C
三点,过坐标原点O的直线
与抛物线交于M、N两点.分别过点C、D
作平行于
轴的直线
、
.(1)求抛物线对应的二次函数的解析式;(2)求证:以ON为直径的圆与直线
相切;(3)求线段MN的长(用
表示),并证明M、N两点到直线
的距离之和等于线段MN的长.
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/4fbddbdfabab4550bfb83c4a46fda115.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/28c38477a2914d08bf56d2768b291972.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/50ae5727b62d4679bdd6542feadbc2ac.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/15a03c0f8234483e834cd3969fb73c58.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/b3bc8f851d394332b24cab8ca34b570e.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/c8e31e3a457f47eda0042ba68a5abd4e.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/02e26592ab7546678bb399f665cd0e99.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/eb5ae2f76bd4437097ad0cc25765eb30.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/02e26592ab7546678bb399f665cd0e99.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/2ba5c2ddc3324d15a776a0334d36274d.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/eb5ae2f76bd4437097ad0cc25765eb30.png)
![](https://img.xkw.com/dksih/QBM/2012/11/9/1571058669568000/1571058675179520/STEM/8073397b2f20477895392153773e3195.png)
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解题方法
4 . 已知抛物线C:
与椭圆E:
的一个交点为
,且E的离心率
.
(1)求抛物线C和椭圆E的方程;
(2)过点A作两条互相垂直的直线AP,AQ,与C的另一交点分别为P,Q,求证:直线PQ过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c9bebea391a1f9956dfcca98d9d1f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1dab74e16403e8131f9f5b2a74f3a84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/075ba8c6fb5ef7288cd3fed425c8e69e.png)
(1)求抛物线C和椭圆E的方程;
(2)过点A作两条互相垂直的直线AP,AQ,与C的另一交点分别为P,Q,求证:直线PQ过定点.
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5 . 已知抛物线
,直线
与
交于
,
两点,且
.
(1)求
的值;
(2)过点
作
的两条切线,切点分别为
,
,证明:直线
过定点;
(3)直线
过
的焦点
,与
交于
,
两点,
在
,
两点处的切线相交于点
,设
,当
时,求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf92a1ba410263d4f68b7e0432b19aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2f3fa679bb55ded25a9b72a8e788cb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](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/b6db6b81e64d6a974fad6d8cd495a1e0.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de4c8ee15e17ecd10bdd25b09812ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
(3)直线
![](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/a0ed1ec316bc54c37c4286c208f55667.png)
![](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/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/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9571c08ff6dfcf72415ab11bf4533cd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90bf910b55f4ce7b6272a9271874434b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3d35f338f9b950b6c9dcd115ad503b2.png)
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6 . 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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7 . 设抛物线
,直线
是抛物线C的准线,且与x轴交于点B,过点B的直线l与抛物线C交于不同的两点M,N,
是不在直线l上的一点,直线
,
分别与准线交于P,Q两点.
(1)求抛物线C的方程;
(2)证明:
:
(3)记
,
的面积分别为
,
,若
,求直线l的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99c6875d552e9fff3c7d655f3a59b166.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aad445f2f16d42d63980353981bdcf48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
(1)求抛物线C的方程;
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2fd1dc3d012bef0ec559463298f5347.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61098066865164289ca1348b53420cd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8503628bad86f077d1f6a4d801314f04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8060a00d925f27135a7baff2d0e9d598.png)
您最近一年使用:0次
2024-04-19更新
|
708次组卷
|
3卷引用:浙江省金华十校2024届高三4月模拟考试数学试卷
名校
解题方法
8 . 已知点F为抛物线C:
的焦点,点
在抛物线C上,且
.
(1)求抛物线C的方程;
(2)若直线l与抛物线C交于M,N两点,设直线AM,AN的斜率分别为
,
,且
,求证:直线l过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9a59fab7b7c1f02252640770c690c0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c2acdc2bc168ae98bf3c398da3ca492.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b59ffc98446c48145631f2e510cce59f.png)
(1)求抛物线C的方程;
(2)若直线l与抛物线C交于M,N两点,设直线AM,AN的斜率分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/151e09f3a383d7811c619eef3a6398cf.png)
您最近一年使用:0次
2024-03-01更新
|
436次组卷
|
2卷引用:浙江省名校协作体2023-2024学年高二下学期2月月考数学试题
名校
解题方法
9 . 已知抛物线
的焦点
关于直线
的对称点为
.
(1)求
的方程;
(2)若
为坐标原点,过焦点
且斜率为1的直线
交
于
两点,求
;
(3)过点
的动直线
交
于不同的
两点,
为线段
上一点,且满足
,证明:点
在某定直线上,并求出该定直线的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82ea1be9b9b6bb12afa7e1ce703d1603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd24f3c4bc9f9a75d4b28630bb630d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18dcc62ada52231d06bb963d00fadc7f.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52586ca2a3b783bc8092415e2d4bf6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2401bc9c26cc3b0b8384c7139bd58fff.png)
(3)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e16d5139682dad62baa20ab740787c.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/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eabe95fc577b9a3f6a715277f12f62a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
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解题方法
10 . 已知抛物线
经过点
中的两个点,准线为
,
为坐标原点.
(1)求准线
的方程;
(2)过点
的直线与抛物线
交于
两点,直线
与
轴交于点
,直线
与
交于点
,过点
作
的垂线,垂足为
,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/0f85fca60a11e1af2bf50138d0e3fe62.png)
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(1)求准线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d56ab70e602f2e2e291df643ab209162.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f09757d013574cf058d5bb944fdf034a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86e203b7c9a6600e0272c58a23733490.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b10a7cf5481c31fae0b86426f8318ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8fc3f8440d6e7a9511507e1bb618eac.png)
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