1 . 已知抛物线
的焦点为
,过点
的直线与抛物线
交于
两点,点
在第一象限,
为坐标原点.
(1)设
为抛物线
上的动点,求
的取值范围;
(2)记
的面积为
的面积为
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/300e33059d3b6a73de31bf90229d4617.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fab8a0cc6504aa4c3a38006f5394b4c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df46779f1906c5e4da98ea451847bf38.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4abb5565ba58a6b7b4c23de8674a498c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4c21cbb1c2bcbcb8391ac5a879f2ae0.png)
您最近一年使用:0次
2023-08-03更新
|
772次组卷
|
5卷引用:贵州省威宁彝族回族苗族自治县第八中学2023届高三数学(理)样卷(二)试题
贵州省威宁彝族回族苗族自治县第八中学2023届高三数学(理)样卷(二)试题(已下线)单元提升卷10 平面解析几何四川省内江市第六中学2023-2024学年高三上学期第一次月考理科数学试题(已下线)第七节 抛物线 第二课时 直线与抛物线的位置关系 B素养提升卷(已下线)热点7-4 抛物线及其应用(6题型+满分技巧+限时检测)
2 . 已知动圆过定点
,且截
轴所得的弦长为4.
(1)求动圆圆心
的轨迹方程;
(2)若点
,过点
的直线交
的轨迹于
两点,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/448c0a5ee776d19ce8e42ac9a5fd27c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
(1)求动圆圆心
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf90f8358fa8cd9bc8c99ea48d9ae1a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7095e1d0e3e6be39a3ac68ad76bd9f26.png)
您最近一年使用:0次
2024-03-21更新
|
720次组卷
|
2卷引用:黑龙江省双鸭山市第三十一中学等校2024届高三第二次模拟数学试题
名校
解题方法
3 . 已知动圆过定点
,且与直线
相切.
(1)求动圆圆心C的轨迹的方程.
(2)设A、B是轨迹C上异于原点O的两个不同点,直线OA和OB的倾斜角分别为
和
,当
,
变化且
为定值
,证明直线AB恒过定点,并求出该定点的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99c6875d552e9fff3c7d655f3a59b166.png)
(1)求动圆圆心C的轨迹的方程.
(2)设A、B是轨迹C上异于原点O的两个不同点,直线OA和OB的倾斜角分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8dc4c63a548b91061528aa11058de75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1ad72d7565699d1ebb741eb0ce12bac.png)
您最近一年使用:0次
2023-11-12更新
|
707次组卷
|
4卷引用:四川省广安第二中学校2023-2024学年高三上学期第二次月考理科数学试题
名校
解题方法
4 . 设抛物线
的准线为l,A、B为抛物线上两动点,
于
,定点
使
有最小值
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/5/f32f8a5e-ab97-41a3-a59b-f0f7f66fbb26.png?resizew=155)
(1)求抛物线的方程;
(2)当
(
且
)时,是否存在一定点T满足
为定值?若存在,求出T的坐标和该定值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebf39f807af4ea7bec39fe04b60566c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67f0ac21a3b9c3e7d11d1547217850b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31be0ce029f4205563bf67a73d55a143.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/5/f32f8a5e-ab97-41a3-a59b-f0f7f66fbb26.png?resizew=155)
(1)求抛物线的方程;
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/941dcd189840d67264beaa7f973b87e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13e87088da41685cc8d433fbbe0e18d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/472393b18c7880e73b40e31fbe2d951c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bc8d143b8678e32e891a2cf552f4682.png)
您最近一年使用:0次
2022-12-04更新
|
1495次组卷
|
10卷引用:广西壮族自治区南宁市第三中学2023届高三模拟数学(理)试题(二)
广西壮族自治区南宁市第三中学2023届高三模拟数学(理)试题(二)四川省成都市第七中学2022-2023学年高二上学期期中数学理科试题四川省成都市成都市第七中学2022-2023学年高二上学期期中数学文科试题四川省成都市树德中学2022-2023学年高二上学期期中考试数学(理)试题云南省大理市下关第一中学教育集团2022~2023学年高二上学期段考(二)数学试题(A卷)云南省下关第一中学2022-2023学年高二上学期段考(二)数学(A卷)试题辽宁省沈阳市东北育才双语学校2022-2023学年高二上学期期末数学试题(已下线)专题04 圆锥曲线经典题型全归纳(2)湖南省长沙市长郡中学2023-2024学年高二上学期期中数学试题(已下线)3.3.2 抛物线的简单几何性质【第三课】“上好三节课,做好三套题“高中数学素养晋级之路
5 . 已知抛物线C:
(
)的焦点为F,直线
与C交于A,B两点,
.
(1)求C的方程;
(2)过A,B作C的两条切线交于点P,设D,E分别是线段PA,PB上的点,且直线DE与C相切,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b35f0b940c8422ef47edc3b7ce55e47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5abd313d4e92a762fb7fb0c1cb65263d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ab466aedd6e176088d8dee7bc3e3aaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49e1bc08cc69d3d8e73b990f1236ed5c.png)
(1)求C的方程;
(2)过A,B作C的两条切线交于点P,设D,E分别是线段PA,PB上的点,且直线DE与C相切,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da715afe51a44ad2f044ccf61c313778.png)
您最近一年使用:0次
名校
解题方法
6 . 已知点
,点B为直线
上的动点,过点B作直线
的垂线l,且线段
的中垂线与l交于点P.
(1)求点P的轨迹
的方程;
(2)设
与x轴交于点M,直线
与
交于点G(异于P),求四边形
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5421a28dc3675ae20190d6090793246e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eefa44964db83759aff6fc8dd7ef8f28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eefa44964db83759aff6fc8dd7ef8f28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2319a01218514917e446dfc807a625ff.png)
(1)求点P的轨迹
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2319a01218514917e446dfc807a625ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fb26d84907c923278ac4626a9d58947.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cca797952ee1e84f43e1a7e80e6ff639.png)
您最近一年使用:0次
2023-02-25更新
|
731次组卷
|
3卷引用:湖南省株洲市2023届高三下学期一模数学试题变式题17-22
(已下线)湖南省株洲市2023届高三下学期一模数学试题变式题17-22福建省厦门市2022-2023学年高二上学期期末考试数学试题四川省成都市第七中学2023-2024学年高二上学期期末复习数学试题(一)
解题方法
7 . 已知抛物线
:
的焦点为
,准线为
,直线
经过点
且与
交于点
、
.
(1)求以
为焦点,坐标轴为对称轴,离心率为
的椭圆的标准方程;
(2)若
,求线段
的中点到
轴的距离;
(3)设
为坐标原点,
为
上的动点,直线
、
分别与准线
交于点
、
.求证:
为常数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.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/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(1)求以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf10573c5ea2b397b8721d1db7f5cc2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.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/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c7906b66906ec5943b3bbd9ce9a47e7.png)
您最近一年使用:0次
8 . 已知动圆过定点
,且与直线
相切,其中
.
(1)求动圆圆心
的轨迹的方程;
(2)设
是轨迹C上异于原点O的两个不同点,直线
和
的倾斜角分别为
和
,当
变化且
为定值
时,证明直线
恒过定点,并求出该定点的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4356a596535d4e905ae47e191940f34f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b64b91d079810d968b9ef63e3284c7af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5abd313d4e92a762fb7fb0c1cb65263d.png)
(1)求动圆圆心
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52586ca2a3b783bc8092415e2d4bf6d.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/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8dc4c63a548b91061528aa11058de75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/798cdca7d20743e0197fe422f09fcbfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
您最近一年使用:0次
2022-11-29更新
|
1582次组卷
|
3卷引用:2005年普通高等学校招生考试数学(理)试题(山东卷)
名校
解题方法
9 . 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)
您最近一年使用:0次
10 . 在①
;②
;③
面积的最小值为8,这三个条件中任选一个,补充在横线上,并解答下列问题.(若选择多个条件作答,则按第一个解答计分)
已知抛物线
的焦点为F,过点F的直线与该抛物线交于A,B两点,O为坐标原点,_____________.
(1)求抛物线的方程;
(2)点C在抛物线上,
的重心G在y轴上,直线
交y轴于点Q(点Q在点F上方).记
的面积分别为
,求T的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59411f424d59bc4de8f2b2e124fdbba9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9520ae34d0893349c8e2a09dbb7943f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
已知抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8516f71467b419293fa27df70bdaed74.png)
(1)求抛物线的方程;
(2)点C在抛物线上,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e75e3c33ea291d63a5fbb161e0806f4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/510317d708a6e4fef916cdf6901bc7b4.png)
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