解题方法
1 . 设A,B为椭圆C:
的短轴端点,P为椭圆上异于A,B的任意一点,D在直线
上.
(1)求直线
,
的斜率的乘积;
(2)证明:
;
(3)过右焦点F作x轴的垂线
,E为
上异于F的任意一点,直线
交C于M,N两点,记直线
,
,
的斜率分别为
,
,
,是否存在
,
,
的某个排列,使得这三个数成等差数列?若存在,加以证明;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cae00bdc6f8b564b6b15b32572c848b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.png)
(1)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6922c5dbeead5d1225b2933e92719f2f.png)
(3)过右焦点F作x轴的垂线
![](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/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21037e170bdbb322558e79c40c00b454.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b46c607b3deac746c0ef3389ad8f65c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c4c865445dda4a59b6d5cb18fd74404.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/dbf434334b09cc0fdd4e86e84e6ceb00.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/dbf434334b09cc0fdd4e86e84e6ceb00.png)
您最近一年使用:0次
2 . 在平面直角坐标系
中,点
,
分别是椭圆
:
的右顶点,上顶点,若
的离心率为
,且
到直线
的距离为
.
(1)求椭圆
的标准方程;
(2)过点
的直线
与椭圆
交于
,
两点,其中点
在第一象限,点
在
轴下方且不在
轴上,设直线
,
的斜率分别为
,
.
(i)求证:
为定值,并求出该定值;
(ii)设直线
与
轴交于点
,求
的面积
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a31eb881e4957090c85757796bc4863d.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f13bf66fc845b115de4ec45b4be0e23.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/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/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
(i)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e1efe96e7776f1b5dfa92c295f8d97d.png)
(ii)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81fa04b7ffc6d376a605688a0bb06537.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
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2卷引用:江苏省南通市如皋中学2024届高三下学期高考适应性考试(三)(3.5模)数学试题
3 . 在平面直角坐标系xOy中,A,B点的坐标分别为
和
,设
的面积为S,内切圆半径为r,当
时,记顶点M的轨迹为曲线C.
(1)求C的方程;
(2)已知点E,F,P,Q在C上,且直线EF与PQ相交于点A,记EF,PQ的斜率分别为
,
.
(i)设EF的中点为G,PQ的中点为H,证明:存在唯一常数
,使得当
时,
;
(ii)若
,当
最大时,求四边形EPFQ的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d06d31c9a2298b02664a86ddd91b1121.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cee9e51b1dd360d8e141e7fa8f4224d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a5e0a51c9e14fb246b0ba0b231c1e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b722e2e83c2d125453ee2d80a5e64d26.png)
(1)求C的方程;
(2)已知点E,F,P,Q在C上,且直线EF与PQ相交于点A,记EF,PQ的斜率分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
(i)设EF的中点为G,PQ的中点为H,证明:存在唯一常数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f132407bf1cc9d1f460d50f1b0547993.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b3fa41635da8da11d6c04287ff7513.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/129fa211eb0cfb3968d38c3c90249842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7560c67fac3a45cf25ec33aa3ca52519.png)
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4 . 已知动圆
与圆
:
和圆
:
都内切,记动圆圆心
的轨迹为
.
(1)求
的方程;
(2)已知圆锥曲线具有如下性质:若圆锥曲线的方程为
,则曲线上一点
处的切线方程为:
.试运用该性质解决以下问题:点
为直线
上一点(
不在
轴上),过点
作
的两条切线
,
,切点分别为
,
.
(ⅰ)证明:
;
(ⅱ)点
关于
轴的对称点为
,直线
交
轴于点
,直线
交曲线
于
,
两点.记
,
的面积分别为
,
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d36f49e116d037d0cf4187d9519ed877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e383fcc122f267043fbafe0972bfb900.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(2)已知圆锥曲线具有如下性质:若圆锥曲线的方程为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/896f353fb463bb5bbfad2d94e8b04971.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23b4f86e48e2b0d63c1865c60ed1e4d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1684f875065fa386c8254eac2e0a3875.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea3680f7d0d96efc5b207c8e9552e21c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/800c5e266b4ad8462a46970f0a232d52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f46b053f98b1d05a2043e94eeaefea87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
(ⅰ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9da27274e261b16a282f76b3c507a9d4.png)
(ⅱ)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ece94d77dd1437041d71dc299aa4ac3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff1e76eb77c8e916d82fe71183881066.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccedd9e7b7f0e54838f58d984519cec6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9eb4830f486b8576b67272d483bf9a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11f4e000e3491c682b81692551ff9b82.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/a38c43d922840f5f7693007beeb62b34.png)
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5 . 如图,矩形
中,
,
,
分别是矩形四条边的中点,设
,
,设直线
与
的交点
在曲线
上.
的方程;
(2)直线
与曲线
交于
,
两点,点
在第一象限,点
在第四象限,且满足直线
与直线
的斜率之积为
,若点
为曲线
的左顶点,且满足
,直线
与
交于
,直线
与
交于
.
①证明:
为定值;
②是否存在常数
,使得四边形
的面积是
面积的
倍?若存在求出
,若不存在说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08e39fda3cda5ddc03b085413f2030aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f17edac849a0691e52146021e05d83e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e56966d92b71ae6ec41ccb88667f5db9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e17a42c1b3c7c8f38e1cb877365b5d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2871d7f054a9313823d6885fd69f071a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba28c45f78fb7643ec9781a800271cc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c7ebdc16bd34f6daddd1a988ab2ac68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)直线
![](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/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f0009063fe00277645aff1be6e32471.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e9f7d1272b7344346b58b660aa260a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/602baac86c2b1668ecdfadc8a5948885.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69837fef2bc60f34cdee393543af5fac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e9f7d1272b7344346b58b660aa260a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8ffe24cf9f327aeb241225ab15ab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f0009063fe00277645aff1be6e32471.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02c07b101a1a118c7558a9e59b13c95c.png)
②是否存在常数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1cbae7bfee1523506ffb27f8adce8554.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d90f8e1d845107aa138d5b6376e54f6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
6 . 已知椭圆
的离心率为
,点
在
上.
(1)求椭圆
的方程;
(2)过点
的直线
交椭圆
于
两点(异于点
),过点
作
轴的垂线与直线
交于点
,设直线
的斜率分别为
.证明:
(i)
为定值;
(ii)直线
过线段
的中点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e90064d012356de1877aa697cd6d6ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60f2a72c6d7780757ab065fb29f47526.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/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d454c82d9e52747563d47b68099249.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5671fb25040a712a49e8c8148d67d300.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
(i)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b69e3f7ddd51215d00661c09cd900d60.png)
(ii)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b1ec05e3cec27677ded7b4aecaa62d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
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7 . 椭圆
:
的离心率为
,圆
:
的周长为
.
(1)求
的方程;
(2)如图,
是
的左焦点,过
的直线交圆O于点M,N,线段
的垂直平分线交C于点P,Q,交
于点A.
(i)证明:四边形
的面积为定值.
(ii)记
,
的面积分别为
,
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/347b68f42934c74e0d759a67613a1da9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1174142f3bba761585b6bc2653009b36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e4be819fc47b2aa19ab2022b3dfeb6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/557a94abe27623e48ee2c729a8352b8c.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)如图,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
(i)证明:四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96198405db093eb43952282a69332f92.png)
(ii)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46377cebc8defe6d156cc7451f5f1278.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f275ea55885ec89eb99fdefec5666c8.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/235f0a6fb218d28383e6f27f2df1f50f.png)
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2024-06-05更新
|
273次组卷
|
2卷引用:辽宁省辽阳市辽阳县辽阳石油化纤公司高级中学2024届高三下学期模拟考试数学试题
解题方法
8 . 已知椭圆
的离心率为
,过点
的直线
与椭圆
交于
两点,当
过坐标原点
时,
.
(1)求椭圆
的方程;
(2)线段
上是否存在定点
,使得直线
与直线
的斜率之积为定值. 若存在, 求出点
的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bec550c01b4f075f22ab67f5e55ed5d.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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d363758a1cc53a47db20561e40638692.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bf25e032b5599ac49383de06e776365.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf702adb116c1e46569eb7050d029f49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
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9 . 数学家加斯帕尔·蒙日在研究圆锥曲线时发现:椭圆
任意两条互相垂直的切线的交点都在以原点O为圆心,
为半径的圆上,这个圆被称为该椭圆的蒙日圆.已知椭圆C:
可以与边长为
的正方形的四条边均相切,它的左、右顶点分别为A,B,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da001dad7941e6c9858637d7b62cec59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec92768b052f6ceb3efc112b7e48625.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b10e8abf8690e4b129466ddb918bcc94.png)
A.![]() |
B.若矩形的四条边均与椭圆C相切,则该矩形面积的最大值为12 |
C.椭圆C的蒙日圆上存在两个点M满足![]() |
D.若椭圆C的切线与C的蒙日圆交于E,F两点,且直线OE,OF的斜率都存在,记为![]() ![]() ![]() |
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解题方法
10 . 已知椭圆
的左右焦点为
,
,P是椭圆C上的动点,
的最大值为8,当
时,
.
(1)求椭圆C的标准方程;
(2)点
,若点M,N在椭圆C上,且直线
,
的斜率乘积为
,线段
的中点G,当直线
与y轴的截距为负数时,求
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.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/560510300ddfc5c893e0d4123bbf9f22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fde69214438f3a720c8094e886eb31dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dccf31e2e06794301c8dff7939521ce.png)
(1)求椭圆C的标准方程;
(2)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1bd72b6e698a649d551a0ec50e3b2a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b09bf6ba623953df55eb869b2b363e39.png)
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