1 . 已知抛物线E的准线方程为:
,过焦点
的直线与抛物线
交于A、B两点,分别过A、B两点作抛物线
的切线,两条切线分别与
轴交于C、D两点,直线CF与抛物线
交于M、N两点,直线DF与抛物线
交于P、Q两点.
的标准方程;
(2)是否存在实数
,使得
恒成立,若存在,求出
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef00713e73b8357cc7900144f5505bc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cda91de2acdaeaa7f4942cf507ffbb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
解题方法
2 . 设抛物线
的方程为
,
为直线
上任意一点;过点
作抛物线
的两条切线MA,MB,切点分别为A,B(A点在第一象限).
(1)当M的坐标为
时,求过M,A,B三点的圆的方程;
(2)求证:直线AB恒过定点;
(3)当m变化时,试探究直线l上是否存在点M,使
为直角三角形,若存在,有几个这样的点,说明理由;若不存在,也请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad696f38c4e395e13ca8ff0beaafef5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)当M的坐标为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bd5f27c5f8a8cda3403c73108dfd30c.png)
(2)求证:直线AB恒过定点;
(3)当m变化时,试探究直线l上是否存在点M,使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a11cb104b04c4e6a1be700e81da279a.png)
您最近一年使用:0次
3 . 两条动直线
和
分别与抛物线
相交于不同于原点的A,B两点,当
的垂心恰是C的焦点时,
.
(1)求p;
(2)若
,弦
中点为P,点
关于直线
的对称点N在抛物线C上,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/766bc42b7ead98238a339bb4dc42bb51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87f8af9ce5d927e6f422de42ead6ffb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c965530523e606200248f2dd720eb5c.png)
(1)求p;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/771a2abef658aab00d81a85ac26d9128.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8547f2b4e89b0ae1445bda02d46f0668.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1ed4c4e8edbd179f3fc38a6653f18c1.png)
您最近一年使用:0次
2024-05-22更新
|
1423次组卷
|
3卷引用:模块7专题6 正交于顶 模型优先练
4 . 已知动圆
过定点
且与直线
相切,记圆心
的轨迹为曲线
.
(1)已知
、
两点的坐标分别为
、
,直线
、
的斜率分别为
、
,证明:
;
(2)若点
、
是轨迹
上的两个动点且
,设线段
的中点为
,圆
与动点
的轨迹
交于不同于
的三点
、
、
,求证:
的重心的横坐标为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5421a28dc3675ae20190d6090793246e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9355031ea0b2dc9cef3777621bc6d38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(1)已知
![](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/879a4007beef22e009248112d664f7c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80991c1f0c963104740e50cfff6f29a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.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/bf209717d3bde602ab96c53d6a43a811.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8198c3b302b3820e86763428eb1e91cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3463ced6030af957f13f9ba05b977c1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfb2356a3833defed220ee1fa481aad2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.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/a0ed1ec316bc54c37c4286c208f55667.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/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcaecf08a22124a457128fb04c9c02bb.png)
您最近一年使用:0次
2024·全国·模拟预测
解题方法
5 . 已知抛物线C:
的焦点为
,过点F的直线与C交于点
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
,C在点A,B处的切线交于点P.
(1)求
的值.
(2)若点D是抛物线C上位于直线AB上方的点,点D处的切线与PA,PB分别交于点M,N,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/049cdb7561122405c1c302079593c987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52408a489fe3167d678aa193bb49cc43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73261d4855082a200e586c01f79e2103.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ceddc345bfa05b7c0c61ec02470188a.png)
(2)若点D是抛物线C上位于直线AB上方的点,点D处的切线与PA,PB分别交于点M,N,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29df87992804e7f1de8af257648a8413.png)
您最近一年使用:0次
2024·全国·模拟预测
解题方法
6 . 已知点
在抛物线
上,点
是抛物线
上的两个动点,直线
与
的倾斜角互补.
(1)求抛物线
的方程和直线
的斜率;
(2)设
的外接圆为圆
,过点
作抛物线
的切线
,证明:直线
与圆
相切.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/115a0c87ac14dbb770c95d74d6e26073.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d454c82d9e52747563d47b68099249.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9763846b1131e1e3e2d741ad95d5bb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
您最近一年使用:0次
7 . 如图,过点
的动直线
交抛物线
于
两点.
,求
的方程;
(2)当直线
变动时,若
不过坐标原点
,过点
分别作(1)中
的切线,且两条切线相交于点
,问:是否存在唯一的直线
,使得
?并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72736d2f88303c634fda02142a12ac53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/222487b7fe2b36e3f6ec4ef42065d2b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a017adb8c3e68b5d5acb23f0760319e8.png)
您最近一年使用:0次
名校
解题方法
8 . 已知抛物线
与椭圆
有公共的焦点.
的标准方程.
(2)如图,过抛物线
的焦点
的直线与抛物线
交于点
,点
,直线AP,BP分别与抛物线
交于点
.证明:
①直线CD过定点;
②
与
的面积之比为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50996be3c0d950d6ef58110948b7fbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82766cfd2b7c59c7fac5b827ae5863b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)如图,过抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.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/de77ee0b176035fd3a89edc2ad957a77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bacfc149ede71417fa599c21b5a84cb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
①直线CD过定点;
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/177678001b2ccde1db8f57fa5e017002.png)
您最近一年使用:0次
2024·全国·模拟预测
解题方法
9 . 已知抛物线
,过点
的直线
与抛物线
相切.
(1)求抛物线
的方程;
(2)若过点
的直线
与
相交于
,
两点,
为
上任意一点且直线
,
与直线
分别交于
,
两点.求证:直线
,
的斜率之积是定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b527ec9f92467b8f24554a2a67ee987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ee80939187a84e1863eeb192a301c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adbba4eaea11a24409e3a22ea70ae468.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)若过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.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/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55aa0a20848c37c1892c567b2315e04.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/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e9f7d1272b7344346b58b660aa260a.png)
您最近一年使用:0次
2024·全国·模拟预测
10 . 已知点
,直线
与抛物线
交于B,C两点(均不同于点A).设直线AB,AC的斜率分别为
,有
.
(1)证明:直线
经过定点.
(2)若B,C两点在
轴的异侧,则存在几条直线
,使
的面积为4?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4212e0f2da912518e8b02a741cc91ad2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42102c1c07562853219ca5918803a27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe91cc3bfc93ef1cc3369fa6756bbd4d.png)
(1)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)若B,C两点在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
您最近一年使用:0次