21-22高二·江苏·课后作业
1 . 已知圆A:(x+2)2+y2=1与定直线l:x=1,且动圆P和圆A外切并与直线l相切,求动圆的圆心P的轨迹方程.
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2022-03-01更新
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4卷引用:3.3.2 抛物线的几何性质
2 . 已知抛物线
的焦点为
,
,
,
为
上不重合的三点.
(1)若
,求
的值;
(2)过
,
两点分别作
的切线
,
,
与
相交于点
,过
,
两点分别作
,
的垂线
,
,
与
相交于点
.
(i)若
,求
面积的最大值;
(ii)若直线
过点
,求点
的轨迹方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f32be681ecfdd5463567671d1ee034d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d856ed2306a7cc3e64a1edfb84627aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae6740760f007759d12ab009808527bb.png)
(2)过
![](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/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.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/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9fce9427c9b17e4d3cda0c3ff3e2e14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f2813ee8f26cca880b6427f5f545d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9fce9427c9b17e4d3cda0c3ff3e2e14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f2813ee8f26cca880b6427f5f545d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(i)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57dfc9d1109fe41145cc892b5702d9fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
(ii)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7a999c36de5c9a9ce876a4a56fa34c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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解题方法
3 . 已知椭圆![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99cd361ce118bca96a731b241a9c587d.png)
与抛物线![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cf1bce17440e83c1b735d2954010a2a.png)
的图象在第一象限交于点P.若椭圆的右顶点为B,且
.
(1)求椭圆C1的离心率;
(2)若椭圆C1的焦距为2,直线l过点B且不与坐标轴垂直.设l与椭圆C1相交于不同于B的另一点D,l与抛物线C2相交于不同于的两点M、N,且
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99cd361ce118bca96a731b241a9c587d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f82eb4ba631d0f50d848aa6e576b379.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cf1bce17440e83c1b735d2954010a2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd8dc6eb286b37746dabe9432639c5f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9eb6436e1649e08b702c54ef2c9166b6.png)
(1)求椭圆C1的离心率;
(2)若椭圆C1的焦距为2,直线l过点B且不与坐标轴垂直.设l与椭圆C1相交于不同于B的另一点D,l与抛物线C2相交于不同于的两点M、N,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cd2c0ca63f220d0eb865365069bb9bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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解题方法
4 . 已知抛物线
的顶点在坐标原点,焦点与圆
的圆心重合,
为
上一动点,点
. 若
的最小值为2.
(1)求抛物线
的标准方程;
(2)过焦点的直线
与抛物线
和圆
自上而下依次交于
四点,且满足
, 求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32b94aff0ebb58510c12aea17a4b1b5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e584f799ea554fc5533925ead4672501.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acea2e2ac2b1786d527f5ee0570d0c83.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)过焦点的直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32e33b939a29ff02fa1df681df1f29c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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解题方法
5 . 已知点
为抛物线
的焦点,点
在抛物线上,且
.
(1)求抛物线
的方程;
(2)过点
分别作两条互相垂直的直线与抛物线
分别交于
与
,记
的面积分别为
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4fdd6886c3766401aff6140a4b38521.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b4216cb8ffd7aa7fb1bc126ffe17b66.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过点
![](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/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/215e3d4ccd3c4153e3e259d2f2f2e3fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3637753af5ce86be9c23a9beb6b5067.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4c21cbb1c2bcbcb8391ac5a879f2ae0.png)
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2023-03-26更新
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652次组卷
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5卷引用:江西省部分学校2023届高三联考数学(文)试题
江西省部分学校2023届高三联考数学(文)试题(已下线)专题14圆锥曲线中的最值、范围、探索问题(已下线)3.3.2 抛物线的简单几何性质(第2课时)(分层作业)(3种题型分类基础练+能力提升练)-【上好课】高二数学同步备课系列(人教A版2019选择性必修第一册)(已下线)2.7.2 抛物线的几何性质(分层练习)-2023-2024学年高二数学同步精品课堂(人教B版2019选择性必修第一册)湖南省常德市汉寿县第一中学2024届高三下学期4月月考数学试题
名校
6 . 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
7 . 在水平桌面上放一只内壁光滑的玻璃水杯,已知水杯内壁为抛物面型(抛物面指抛物线绕其对称轴旋转
所得到的面),抛物面的轴截面是如图所示的抛物线.现有一些长短不一、质地均匀的细直金属棒,其长度均不小于抛物线通径的长度(通径是过抛物线焦点,且与抛物线的对称轴垂直的直线被抛物线截得的弦),若将这些细直金属棒,随意丢入该水杯中,实验发现:当细棒重心最低时,达到静止状态,此时细棒交汇于一点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/23/b65dbcb7-dc60-447f-9218-3d0e46d8e8dd.png?resizew=281)
(1)请结合你学过的数学知识,猜想细棒交汇点的位置;
(2)以玻璃水杯内壁轴截面的抛物线顶点为原点,建立如图所示直角坐标系.设玻璃水杯内壁轴截面的抛物线方程为
,将细直金属棒视为抛物线的弦
,且弦
长度为
,以细直金属棒的中点为其重心,请从数学角度解释上述实验现象.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54e7a123c9cc0e058db28841fb0edcf3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/23/b65dbcb7-dc60-447f-9218-3d0e46d8e8dd.png?resizew=281)
(1)请结合你学过的数学知识,猜想细棒交汇点的位置;
(2)以玻璃水杯内壁轴截面的抛物线顶点为原点,建立如图所示直角坐标系.设玻璃水杯内壁轴截面的抛物线方程为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b35f0b940c8422ef47edc3b7ce55e47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d73879aefd41beb91cc808904276b1d.png)
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8 . 如图,已知点
是焦点为
的抛物线
:
上一点,
,
是抛物线
上异于
的两点,且直线
,
的倾斜角互补,若直线
的斜率为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/2/38675591-810e-4a8d-a961-fd6e89ba6ea6.png?resizew=198)
(1)证明:直线
的斜率为定值;
(2)在
中,记
,
,求
最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2007972af3341f27fbc32ce62dfce5e2.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/7089148c36cb3c39af71de653756396a.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2781714853ddd3675560abfaa967242.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/2/38675591-810e-4a8d-a961-fd6e89ba6ea6.png?resizew=198)
(1)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(2)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/004104bafb5f30338123d4ea2b7fedde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7cd0e6d684f1983034c305af2f24cc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06433d26939d333c62823d6113f98177.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d94ff61f05b1a88f9f9f785441bfcc92.png)
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2022-11-01更新
|
990次组卷
|
4卷引用:广西普通高中2023届高三摸底测试数学(理)试题
解题方法
9 . 已知抛物线
上一点
的纵坐标为
,点
到焦点
的距离为
.过点
做两条互相垂直的弦
、
,设弦
、
的中点分别为
.
(1)求抛物线
的方程;
(2)过焦点
作
,且垂足为
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df82de529d77b7f7a2ad99c5aeb920de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6670479a0083dd2dfd5ad55b47b1ab6.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)过焦点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5346de3855252a225dbe3677ff22e65a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/529a9b0d2b9baa16e3f1aab8b946f270.png)
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解题方法
10 . 已知抛物线
的焦点到准线间的距离为2,且点
抛物线C上.
(1)求m的值;
(2)若直线l与抛物线C交于A,B两点,且
,
于点D,
,求DQ的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e23b74ed51912d236727c0ee5e3e64.png)
(1)求m的值;
(2)若直线l与抛物线C交于A,B两点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83640592853a53872d7af69c0cffc1bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/545e18836bc7fee22f8f813a6f525d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a291e0d0e2f620a15826a1aa3c04bc37.png)
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