名校
1 . 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)
您最近一年使用:0次
2020-01-31更新
|
222次组卷
|
4卷引用: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
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重庆市南开中学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20210304-003(已下线)【新东方】高中数学20210323-002【高二上】
2 . 已知抛物线
:
与双曲线
:
相交于点
.
(1)若
,求抛物线
的准线方程;
(2)记直线l:
与
、
分别切于点M、N,当p变化时,求证:
的面积为定值,并求出该定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac3a4f9f9ae0a23a09a780f3073d132a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b79ca81f286d8aeed52f91ee13ce0d8.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8faa47316e8d6c9631775a9d170bd26e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)记直线l:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15fb18163df0690365a0d2e7ee88f5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2ca3d56b93b5f218eaebb87045cd2b.png)
您最近一年使用:0次
3 . 在平面直角坐标系xOy中,过点
的直线
与抛物线
交于M,N两点
在第一象限).
(1)当
时,求直线
的方程;
(2)若三角形OMN的外接圆与曲线
交于点
(异于点O,M,N),
(i)证明:△MND的重心的纵坐标为定值,并求出此定值;
(ii)求凸四边形OMDN的面积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2ba2238d6afe0187534155dd9ac48c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb4dd4670828f75bc573b52cdd02e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9192616790cac39e605075941ae408c5.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04e28faf289d327e5b67e1da974a7b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)若三角形OMN的外接圆与曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(i)证明:△MND的重心的纵坐标为定值,并求出此定值;
(ii)求凸四边形OMDN的面积的取值范围.
您最近一年使用:0次
2024-04-23更新
|
1627次组卷
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3卷引用:重庆市南开中学校2023-2024学年高二下学期3月定时练习数学试题
4 . 已知抛物线
的准线
交
轴于
,过
作斜率为
的直线
交
于
,过
作斜率为
的直线
交
于
.
(1)若抛物线的焦点
,判断直线
与以
为直径的圆的位置关系,并证明;
(2)若
三点共线,
①证明:
为定值;
②求直线
与
夹角
的余弦值的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f32e577ae1f4449efbd64c1199efe7a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a436db19eb954d31075d5398f1b92ecd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f96c1f9f575ea0fd1487c9f4bb7745af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52041559f8fee18bfa3e2e2ac07c3bfa.png)
(1)若抛物线的焦点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b0d4e207a27fc83c2d7bf247841a788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/939658de45d1e76de0a85e1c50d29e12.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29212e455dd6df5379ae67f379a32158.png)
②求直线
![](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/c24095e409b025db711f14be783a406c.png)
您最近一年使用:0次
2023-12-22更新
|
574次组卷
|
2卷引用:重庆市沙坪坝区重庆一中2024届高三上学期12月月考数学试题
名校
解题方法
5 . 已知
是抛物线
的焦点,抛物线上点A满足AF垂直于x轴,且
.
(1)求抛物线C的标准方程;
(2)
是该抛物线上的两点,
,求线段
的中点到
轴的距离;
(3)已知点
,直线
过点
与抛物线交于
,
两个不同的点
均与点H不重合
,设直线
,
的斜率分别为
,
,求证:
为定值.
![](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/d653eb1ea0106e5bb146dae2e4e0e34f.png)
(1)求抛物线C的标准方程;
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab609a6574633ebabcff3e73fa862081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ea647b776699a17f47fae1f25dc158.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
(3)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c627ad9fd5be5001321b0ca8ce941e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f28c6ce14821fdbe53aca895e52ab1e7.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/fd995178601c2ad7b40f973d268c7bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04582116cd765fcc5a52f44279ad6c94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0a8588dc60a543ad70d6bc0d263dbd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eef1f7b9adab87736321e30949a4d668.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/b4757181824e15e0f21e5bdd55448783.png)
您最近一年使用:0次
名校
解题方法
6 . 已知椭圆
的上、下顶点分别为A,B,O为椭圆的中心,D是线段OB的中点.直线
,动点T到直线m的距离与T到点
的距离相等.设动点T的轨迹为
.
(1)求
的方程;
(2)过点D作斜率为
的直线l,交
于M,N,直线
分别交
于P,Q两点(P,Q均不同于点A),设直线
的斜率为
,求证:
是定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a0ad41722598d9fd00138a0f781b5df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28c2debb1285d2bb3b7907bece265c01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0436039bcb24dab81c2664575dbad83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(2)过点D作斜率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e2fd6df03fd6e945e2c1cb44bca014b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d42cb68c5c877a455ba7ac0a6b6a651.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2a3f348a942d468f0d77c0dfbb41d87.png)
您最近一年使用:0次
2024高三·全国·专题练习
7 . 已知O为坐标原点,抛物线
,过点
的直线交抛物线于A,B两点,
.
(1)求抛物线C的方程;
(2)若点
,连接AD,BD,证明:
;
(3)已知圆G以G为圆心,1为半径,过A作圆G的两条切线,与y轴分别交于点M,N且M,N位于x轴两侧,求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82a1eb1babf5a1eb0f95f3af5c53fcd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15aab800f41f7a47bde139498e19d294.png)
(1)求抛物线C的方程;
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33ab82c33e6c1f8b73628fa78e6868b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9c210f0bb0b964d8b059fdea307a90f.png)
(3)已知圆G以G为圆心,1为半径,过A作圆G的两条切线,与y轴分别交于点M,N且M,N位于x轴两侧,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98013a5042685a1db94249e70c62c09a.png)
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2024-04-08更新
|
2058次组卷
|
5卷引用:重庆市渝北中学校2023-2024学年高三下学期5月月考质量监测数学试题
重庆市渝北中学校2023-2024学年高三下学期5月月考质量监测数学试题(已下线)2024年普通高等学校招生全国统一考试数学文科猜题卷(七)湖北省黄冈市浠水县第一中学2024届高三下学期第三次高考模拟数学试题(已下线)数学(新高考卷03,新题型结构)(已下线)压轴题02圆锥曲线压轴题17题型汇总-2
8 . 已知抛物线
与直线
相交于A、B两点,O为坐标原点.
(1)求证:
;
(2)当
时,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e8953ded144195804384dcb494d5e2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/303094682b317daea83be885d1c7ff4b.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3825ccc273ef9a672a606432d165b866.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45b188ace13b36790a423b141e0eefa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
2023-11-10更新
|
598次组卷
|
4卷引用:重庆市南开中学校2023-2024学年高二上学期期中数学试题
重庆市南开中学校2023-2024学年高二上学期期中数学试题吉林省长春市第六中学2023-2024学年高二上学期1月期末考试数学试题(已下线)3.3.2 抛物线的几何性质(5大题型)-【题型分类归纳】2023-2024学年高二数学同步讲与练(苏教版2019选择性必修第一册)山东省烟台爱华高级中学2023-2024学年高二上学期期末模拟数学试题(一)B卷
名校
解题方法
9 . 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.
您最近一年使用:0次
2023-05-20更新
|
411次组卷
|
2卷引用:重庆市第八中学校2023-2024学年高二下学期入学适应性训练数学试题
解题方法
10 . 已知抛物线
,直线
与抛物线相交于
两点.
(1)证明:
为定值;
(2)当
时,直线![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcdff227aecbb759809b685fd984724f.png)
与抛物线相交于
两点,其中
,
.是否存在实数
,使得经过
两点的直线斜率为2,若存在求线段
的长度,若不存在说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdf017dcbadc8f6b9cc5152815dc9d3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e3a1467ecf286e3cadaf5aa006606f2.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a85af6eaff9a4e3e9af4e9c1f4f7b996.png)
(2)当
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