1 . 已知抛物线
上一点
的纵坐标为4,点
到焦点
的距离为5,过点
做两条互相垂直的弦
、
.
(1)求抛物线
的方程.
(2)求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8ddb5abbec5cc6c4f77fb23e2515f46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/532fb9d8a0715b513e9e3f144ea264e9.png)
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解题方法
2 . 已知抛物线关于
轴对称,顶点在原点,且经过点
,动直线
不经过点
、与
相交于
、
两点,且直线
和
的斜率之积等于3.
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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3 . 已知曲线
是平面内到定点
与到定直线
的距离之和等于
的点的轨迹,若点
在
上,对给定的点
,用
表示
的最小值,则
的最小值为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed54d1cbc27a97aa4ea81f355b773bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/314bf8f0ce875b4dd66ba692fd6a75c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38888ab1beb571a563ea30511698e5a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/309bfced5fe755274e618762ead32061.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a518f9181182af6c2c0867054b580c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/309bfced5fe755274e618762ead32061.png)
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4 . 在平面直角坐标系
中,点
,点A为动点,以线段
为直径的圆与
轴相切,记A的轨迹为
,直线
交
于另一点B.
(1)求
的方程;
(2)
的外接圆交
于点
(不与O,A,B重合),依次连接O,A,C,B构成凸四边形
,记其面积为
.证明:
的重心在定直线上;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f449cadb49859b80c31ef1f68bfe81b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ea16ceca816f7d3d50650af141baf42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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5 . 已知抛物线的焦点为F,原点为O,过F作倾斜角为
的直线l交抛物线C于A,B两点.
(1)过A点作抛物线准线的垂线,垂足为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d71fe246270d1277f9eb2bf15af22e83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47153fdd73c0661fa460130082e30929.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56bbafa889700c6764ebc7fc1a42cd91.png)
(2)当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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6 . 已知方程![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d73e41be08cb1a0247f56b020905b922.png)
(1)试证:不论
如何变化,方程都表示顶点在同一椭圆上的抛物线
(2)
为何值时,该抛物线在直线
上截得的弦最长?并求出此弦长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d73e41be08cb1a0247f56b020905b922.png)
(1)试证:不论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/914b8992c23d0835e27dced0db075ad0.png)
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解题方法
7 . 抛物线
的焦点到准线的距离等于椭圆
的短轴长.
(1)求抛物线
的方程;
(2)设
是抛物线
上位于第一象限的一点,过
作
(其中
)的两条切线,分别交抛物线
于点
,过原点作直线
的垂线,垂足为
,证明点
在定圆上,并求定圆方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f80080fac68745fe783b879cccb6140.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad49499d3c2950c670d56cb6d2b4c3c6.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58ac4e21ffc090dda946c0f24b1008f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d233b47191c89758d648790799f9f604.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f7910d0e12b74383a4914078b562038.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.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/acc290b44635265137fdf13146b6a6d9.png)
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8 . 古希腊的几何学家用一个不垂直于圆锥的轴的平面去截一个圆锥,将所截得的不同的截口曲线统称为圆锥曲线如图所示的圆锥中,AB为底面圆的直径,M为PB中点,某同学用平行于母线PA且过点M的平面去截圆锥,所得截口曲线为抛物线.若该圆锥的高
,底面半径
,则该抛物线焦点到准线的距离为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae890f9e8b32aa53a54158f24f4a87bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9774f83067ed956a551bc41adcce0469.png)
A.2 | B.3 | C.![]() | D.![]() |
您最近一年使用:0次
2024-03-13更新
|
255次组卷
|
4卷引用:河北省石家庄精英中学2023-2024学年高二上学期期末数学试题
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解题方法
9 . 已知抛物线
:
上的点
到焦点
的距离为
.
(1)求抛物线
的方程;
(2)过抛物线上一点
(异于坐标原点)作切线
,过
作直线
,
交抛物线于
,
两点.记直线
,
的斜率分别为
,
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/614ae631e46a92c7f2679dbcf9426807.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e475f27e0d32f098389d6257801a0663.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2be8d8438bffea49d662a15f31b98a42.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过抛物线上一点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a23dff476ddce008f686a26be4adbef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.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/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.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/0fa3b84e2ec9190db6517d8043dc5b32.png)
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10 . 根据抛物线的光学性质可知,从抛物线的焦点发出的光线经该抛物线反射后与对称轴平行.已知抛物线C:
,如图,点F为C的焦点,过F的光线经拋物线反射后分别过点
,
.
(1)求C的方程;
(2)设点
,若过点
的直线与C交于R,T两点,求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3764ba3aa0a241787f4661026bb14053.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdfdbde5e45f33dd3eab3fcf45e796ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df7790bf081318706b9962b6aba7784f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/10/c80cf00c-4b20-441f-9719-061653c773d5.png?resizew=121)
(1)求C的方程;
(2)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b3126e09b4d8e177e618d0169dd98d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44bdbc62c678a23e5e7fc8c34ffbf257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce634840a967c7a7444c0b079538320.png)
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