1 . 求满足下列条件的曲线方程:
(1)已知双曲线的焦点在x轴上,离心率为
,且经过点
;
(2)若动点P在
上移动,求点P与点
连线的中点的轨迹方程.
(1)已知双曲线的焦点在x轴上,离心率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caa585b9257ed0798213a9ae9b87d291.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c9524efddc807d6c9ebf37b1cf37628.png)
(2)若动点P在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2751c5819303b8e60add2356bd7c808b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/203b64e2a9e4ac8bdfb1b541597f7119.png)
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2 . 已知
是抛物线
上任意一点,且
到
的焦点
的最短距离为
.直线
与
交于
两点,与抛物线
交于
两点,其中点
在第一象限,点
在第四象限.
(1)求抛物线
的方程.
(2)证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d9c396aa08378615623bc019d6a2831.png)
(3)设
的面积分别为
,其中
为坐标原点,若
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f80080fac68745fe783b879cccb6140.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4b8503f4706b8321e4e79a87eadea84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e3a1467ecf286e3cadaf5aa006606f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03d7b953c4a7f883fbad5e6cfbbff9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44a977bb284c4faf6abd81a40c3f9f8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098a3e7d1f1890863b7483a98b618119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbdb21011ea821b91d539cb763aac649.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d9c396aa08378615623bc019d6a2831.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9943e86f56a8b70694ebe13b0b0c0189.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3637753af5ce86be9c23a9beb6b5067.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af725b608acc47c1b8a8834b7c31c65d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235f0a6fb218d28383e6f27f2df1f50f.png)
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3 . 已知点
,动点P到y轴的距离为d,且
,记点P的轨迹为曲线C.
(1)求C的方程;
(2)若
,
是C上不同的两点,点A在第一象限,直线
的斜率为k,且
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99296bab1b42898e7ca336a822510258.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/791557dfabd573162b8b424092e2274c.png)
(1)求C的方程;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/921502954d8f4c6e58a95487018a8a04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16456252f5180e3d3619247984d5b26e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d432226fb06dd15f26d72d9aee824b0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
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4 . 已知直线
经过抛物线
的焦点F,且l与C相交于A,B两点,则下列结论中正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ae1fa1c1e342fc48868d130b0cdb3bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df40ba57bb5819b4aaa38d514500052.png)
A.![]() |
B.![]() |
C.![]() |
D.以![]() |
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3卷引用:云南省红河州弥勒市第一中学2023-2024学年高二下学期期中检测数学试题
5 . 已知抛物线
,其焦点到准线的距离为
,斜率为
的直线
与
的交点为
两点,与
轴的交点为
.
(1)求抛物线
的标准方程;
(2)若
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df40ba57bb5819b4aaa38d514500052.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4b8503f4706b8321e4e79a87eadea84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4b8503f4706b8321e4e79a87eadea84.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/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c077d6707bbc50cbe9eb0a6e7cacbfc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4dfec890cdfdda355e19463f3be813.png)
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6 . 已知两个定点
是坐标系原点,
轴于点
是线段
上任意一点,
轴于点
于点
与
相交于点
,则点
与点
之间的距离的最大值和最小值的和等于_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44a12b3ab4ef56755e53ae3b9bd1dd52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b45db8dd8768994af51206565379fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03ab3e9f17b2020538343506e7c2b75d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa8f1243f51b36c405e461705dd1d555.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c75415035b34c75b7096953cf7771fc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c16f0cf7a7f1e82a6364ad66720e45e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be2e2c0d4ac2bd79f6cea7a9b1a50662.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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7 . 已知O为坐标原点,抛物线C:
的准线方程为
,过焦点F的直线l交抛物线C于A,B两点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef00713e73b8357cc7900144f5505bc6.png)
A.若![]() ![]() |
B.若![]() |
C.![]() |
D.![]() |
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8 . 已知拋物线
,过其准线上的点
作
的两条切线,切点分别为
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9158f21b372fd0390fab040ad65c586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc9aa334c8a364cbc597127d60b2b51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab609a6574633ebabcff3e73fa862081.png)
A.抛物线![]() ![]() | B.![]() |
C.直线![]() ![]() | D.直线![]() ![]() |
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9 . 已知
为坐标原点,点
在抛物线![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39dbdab996556fcf8b6e1acc0aa2e3f8.png)
上,过点
的直线交抛物线
于
两点,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bd056ad7b4674fe46f04643fe175538.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39dbdab996556fcf8b6e1acc0aa2e3f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b75ed10db4b9747a4b6d865b774f6b6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81b54b9cf95418bc3dce6e4c698b9907.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
A.抛物线![]() ![]() | B.直线![]() ![]() |
C.![]() ![]() | D.![]() |
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10 . 已知抛物线
:
的焦点为
,点
在
上,且
(
为坐标原点).
(1)求抛物线
的标准方程;
(2)过点
的直线与抛物线
交于点A,B两点,若
为定值,求实数
的值.
![](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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c5718c4cf68779669eb41ff652b7e8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25d84ed7869e0d04058f98ffd902f4ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.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/698b978181ab242af2e961cbc508dfed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26e5109b55b3b96af00c792527c77cca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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