2024高三·全国·专题练习
解题方法
1 . 已知点
,
.以坐标原点O为对称中心且焦点在y轴上的椭圆Ω的离心率为
,过点A且不与坐标轴垂直的直线l与椭圆Ω交于C,D两点,x轴恰平分
,则椭圆Ω的标准方程为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8748dc55e2f45bc37fc4d84d7310f79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/316ba5cbb31299d683ac6c7dd795db85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76bc8e50eeccced1d7f3bc52e09a7d03.png)
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2 . 已知椭圆
的离心率为
,直线
与椭圆C相交于两点M,N,且
.
(1)求C的方程;
(2)若点
,直线
与椭圆C交于两点B,D,且与x轴交于点T.连接
和
.从下列三个条件中选取一个作为条件,探究直线l是否过定点,如是,请求出,如果不是,请说明理由.
①点B关于x轴的对称点在直线
上;
②若直线
与直线
的倾斜角分别为
,
,且满足
;
③B,D两点不在x轴上,设
和
的面积分别为
和
,且
.
注:如果选择多个条件分别解答,按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e72f9c2da79c2d5cea3a25540b48b49.png)
(1)求C的方程;
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a072f7e07dcefc3db2a0443214eedaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fc5bd66dd6d5e09ff0893a938aed56e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
①点B关于x轴的对称点在直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
②若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.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/c1dbab7580d1671f4d5fa8cfa5c5e981.png)
③B,D两点不在x轴上,设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf6b4539674e585ecd20d20e8974b611.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4088a93100186f0510eba81f962ba90e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39931f7165558c636da27ca0f5765182.png)
注:如果选择多个条件分别解答,按第一个解答计分.
您最近一年使用:0次
解题方法
3 . 已知椭圆![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80fbe5e9b0abbbaeccc641cee1b637c4.png)
.
(1)已知椭圆
的离心率为
,求椭圆
的标准方程;
(2)已知直线
过椭圆
的右焦点且垂直于
轴,记
与
的交点分别为A、B,A、 B两点关于y轴的对称点分别为
、
,若四边形
是正方形,求正方形
的内切圆的方程;
(3)设О为坐标原点,P、Q两点都在椭圆
上,若
是等腰直角三角形,其中
是直角,点Р在第一象限,且O、P、Q三点按顺时针方向排列,求b的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80fbe5e9b0abbbaeccc641cee1b637c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40c1fb9f8b59508b1b58180c899d1787.png)
(1)已知椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953cec61ac602ce5eb59b7912352179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdcfe69b939fd1c271747fe9d37ccdf9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdcfe69b939fd1c271747fe9d37ccdf9.png)
(3)设О为坐标原点,P、Q两点都在椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea1f0417d8269f01d8e0bc1a8756e2ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93c3509f3ade93b697d3ccd055d345e4.png)
您最近一年使用:0次
解题方法
4 . 如图,双曲线的中心在原点,焦距为
,左、右顶点分别为A,B,曲线C是以双曲线的实轴为长轴,虚轴为短轴,且离心率为
的椭圆,设P在第一象限且在双曲线上,直线BP交椭圆于点M,直线AP与椭圆交于另一点N.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/12/76fd7703-ba69-43fb-bb0c-ae42a26794b6.png?resizew=223)
(1)求椭圆及双曲线的标准方程;
(2)设MN与x轴交于点T,是否存在点P使得
(其中
,
为点P,T的横坐标),若存在,求出P点的坐标,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbff61fe9d4e93d7cc338489d1c99c40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/12/76fd7703-ba69-43fb-bb0c-ae42a26794b6.png?resizew=223)
(1)求椭圆及双曲线的标准方程;
(2)设MN与x轴交于点T,是否存在点P使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75aa6900b27c1c0dc4aa115e7b8de864.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2297930d54a0452220d963bfef6a616a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d086d58ae2d07117f8bdeb1f63d1f9dc.png)
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5 . 阅读材料:
(一)极点与极线的代数定义;已知圆锥曲线G:
,则称点P(
,
)和直线l:
是圆锥曲线G的一对极点和极线.事实上,在圆锥曲线方程中,以
替换
,以
替换x(另一变量y也是如此),即可得到点P(
,
)对应的极线方程.特别地,对于椭圆
,与点P(
,
)对应的极线方程为
;对于双曲线
,与点P(
,
)对应的极线方程为
;对于抛物线
,与点P(
,
)对应的极线方程为
.即对于确定的圆锥曲线,每一对极点与极线是一一对应的关系.
(二)极点与极线的基本性质、定理
①当P在圆锥曲线G上时,其极线l是曲线G在点P处的切线;
②当P在G外时,其极线l是曲线G从点P所引两条切线的切点所确定的直线(即切点弦所在直线);
③当P在G内时,其极线l是曲线G过点P的割线两端点处的切线交点的轨迹.
结合阅读材料回答下面的问题:
(1)已知椭圆C:
经过点P(4,0),离心率是
,求椭圆C的方程并写出与点P对应的极线方程;
(2)已知Q是直线l:
上的一个动点,过点Q向(1)中椭圆C引两条切线,切点分别为M,N,是否存在定点T恒在直线MN上,若存在,当
时,求直线MN的方程;若不存在,请说明理由.
(一)极点与极线的代数定义;已知圆锥曲线G:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/725a9d6a7c0cd596ece7f4c888b40510.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8112f9185c7d48b015d9cd0525616b31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae8f8c1244657aec3fe29890c4681414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d902a46e5e698f08b6e82c887cee9647.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0a89e3c30f6e4d4c5db4378b05d987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1677a6b74dc0182296d2fb525ce564b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8112f9185c7d48b015d9cd0525616b31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8112f9185c7d48b015d9cd0525616b31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a9656735f55e5de465e5667ba578d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc49a26cb0c64cea942bff447becfdbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8112f9185c7d48b015d9cd0525616b31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7555cd38f338e8758f5f73e10c08dc0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fc58c62444bf42a25289c45425a00f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8112f9185c7d48b015d9cd0525616b31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ba380c2fbc3e6c45bb1dc28a15e219a.png)
(二)极点与极线的基本性质、定理
①当P在圆锥曲线G上时,其极线l是曲线G在点P处的切线;
②当P在G外时,其极线l是曲线G从点P所引两条切线的切点所确定的直线(即切点弦所在直线);
③当P在G内时,其极线l是曲线G过点P的割线两端点处的切线交点的轨迹.
结合阅读材料回答下面的问题:
(1)已知椭圆C:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
(2)已知Q是直线l:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0bbd45a300cd506c9d2bbf8f6ac3498.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/940f1047bde206726ab05cfd6785067d.png)
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2023-02-19更新
|
1354次组卷
|
6卷引用:第五篇 向量与几何 专题5 调和点列 微点4 调和点列综合训练
(已下线)第五篇 向量与几何 专题5 调和点列 微点4 调和点列综合训练(已下线)重难点突破18 定比点差法、齐次化、极点极线问题、蝴蝶问题(四大题型)(已下线)专题22 新高考新题型第19题新定义压轴解答题归纳(9大题型)(练习)贵州省贵阳市普通中学2022-2023学年高二上学期期末监测考试数学试题辽宁省名校联盟2023-2024学年高二下学期4月联合考试数学试卷河南省信阳市新县高级中学2024届高三4月适应性考试数学试题
解题方法
6 . 法国数学家加斯帕尔·蒙日被誉为画法几何之父.他在研究椭圆切线问题时发现了一个有趣的重要结论:一椭圆的任两条互相垂直的切线交点的轨迹是一个圆,尊称为蒙日圆,且蒙日圆的圆心是该椭圆的中心,半径为该椭圆的长半轴与短半轴平方和的算术平方根.已知在椭圆
中,离心率
,左、右焦点分别是
、
,上顶点为Q,且
,O为坐标原点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/15/96a72500-b660-4939-ac97-3e0506c1e79d.png?resizew=215)
(1)求椭圆C的方程,并请直接写出椭圆C的蒙日圆的方程;
(2)设P是椭圆C外一动点(不在坐标轴上),过P作椭圆C的两条切线,过P作x轴的垂线,垂足H,若两切线斜率都存在且斜率之积为
,求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5c7316976a221c051a2c14df80b1347.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8180a6fc0cc7f491751b27f75d338457.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/15/96a72500-b660-4939-ac97-3e0506c1e79d.png?resizew=215)
(1)求椭圆C的方程,并请直接写出椭圆C的蒙日圆的方程;
(2)设P是椭圆C外一动点(不在坐标轴上),过P作椭圆C的两条切线,过P作x轴的垂线,垂足H,若两切线斜率都存在且斜率之积为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3389f53711264b0acba3ba6019f8b908.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df9babb5b5ef3f30621833d88ab6e0d1.png)
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7 . 已知椭圆
的中心在坐标原点
,焦点在
轴上,左、右焦点分别为
、
,离心率
,短轴长为2,.
(1)求椭圆
的标准方程;
(2)设过
且斜率不为零的直线
与椭圆
交于
、
两点,过
作直线
的垂线,垂足为
,证明:直线
恒过一定点,并求出该定点的坐标;
(3)过点
作另一直线
,与椭圆分别交于
、
两点,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/075ba8c6fb5ef7288cd3fed425c8e69e.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)设过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9ea9f5967c96e60f03f332fd792cd21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff9d2abf13c2842f58654abf73c6b4ee.png)
(3)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0e1ba8ef888dfe9a639dddd38d6d603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9fce9427c9b17e4d3cda0c3ff3e2e14.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/17ae93479f71c86ffa37131326cfd993.png)
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2021-12-10更新
|
1117次组卷
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3卷引用:易错点12 圆锥曲线-备战2022年高考数学考试易错题(新高考专用)
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