名校
解题方法
1 . 已知定点
,
,动点P满足
,记动点P的轨迹为
.
(1)求动点P的轨迹
的方程;
(2)已知动直线l的方程为
,其中
,若动直线l与曲线
相交于M、N两点,且点M关于x轴的对称点为点Q,求证:直线QN经过一定点,并求出该定点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f4c1016bf87b416cff0f3fa79d3ef9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/513eafd10fa1ec0196562865517e0b09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/371575b97471ba19edd7b65aad138cac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(1)求动点P的轨迹
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)已知动直线l的方程为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1d0aafd52e26c241c46d0206f42f415.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37b97b295f88972ba1c7e3cefda0885d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
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名校
解题方法
2 . 已知椭圆
:
的左、右焦点分别为
,
,椭圆
的离心率为
,椭圆
上的一点P满足
轴,且|
.
(1)求椭圆
的标准方程:
(2)已知点A为椭圆
的左顶点,若点B,C为椭圆
上异于点A的动点,设直线AB,AC的斜率分别为kAB,kAC,且
,求证:直线BC过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.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/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0803835d6f594a60bd16c823e3ad2cd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf77228c16fdebd33f449e3975c48bc8.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)已知点A为椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54ecdff40dc8025ae9733c61b2ecf9e8.png)
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解题方法
3 . 已知椭圆
的左、右顶点分别为A,B,点
,连接
交椭圆C于点M、N,
为直角三角形,且
.
(1)求椭圆的标准方程;
(2)设直线l与椭圆C交于D、E两点,若
,求证:直线l过定点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5c10f14aae6fb21e047ecb39cdf40c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/790ef3382b1c731f2885eecfd92c2a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2c712d023a4e1ade5513a66882b3478.png)
(1)求椭圆的标准方程;
(2)设直线l与椭圆C交于D、E两点,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5dfc4856e86d8301741cbe2c1e9ce9ec.png)
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2022-03-16更新
|
852次组卷
|
3卷引用:四川省凉山州宁南中学2021-2022学年高二下学期第一次月考数学(理)试题
名校
解题方法
4 . 已知椭圆C的方程为
,斜率为
的直线l与C相交于M、N两点.
(1)若G为MN的中点,且
,求椭圆C的方程;
(2)在(1)的条件下,若P是椭圆C的左顶点,
,F是椭圆的左焦点.
①证明直线l恒过一个顶点,并求出该定点坐标;
②若点F在以MN为直径的圆内,求k的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b060841938313042cc2963204e7bd17f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2799abb64fd7bfce9dfa7228aa460564.png)
(1)若G为MN的中点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a89b87431e1683cabef6a252ec708b0.png)
(2)在(1)的条件下,若P是椭圆C的左顶点,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35e7563ecdfb5e68d2b1ddc3230a901a.png)
①证明直线l恒过一个顶点,并求出该定点坐标;
②若点F在以MN为直径的圆内,求k的取值范围.
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解题方法
5 . 已知椭圆
:
(
)的短半轴长为
,离心率为
.
(1)求椭圆
的方程;
(2)若直线
:
与椭圆
相交于
,
两点(
,
不是左右顶点),且以
为直径的圆过椭圆
的左顶点
,求证:直线
过定点,并求出该定点的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15b256345d7109e081b7c895591e995d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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2022-03-05更新
|
657次组卷
|
3卷引用:安徽省六安市第一中学2021-2022学年高二下学期开学考试数学试题
6 . 已知
,
两点分别在x轴和y轴上运动,且
,若动点G满足
,动点G的轨迹为E.
(1)求E的方程;
(2)已知不垂直于x轴的直线l与轨迹E交于不同的A、B两点,
总满足
,证明:直线l过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cd357b09ef893323574d0173152be6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fa1b0e14ddc29c1459b40dd8e4cb173.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53b75f71a396f2910be554b1c71f51a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ddffc98f53b4c0528a84524d7716811.png)
(1)求E的方程;
(2)已知不垂直于x轴的直线l与轨迹E交于不同的A、B两点,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b514eaf1d034ac68a08365a6d656f4d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b99bdd50e548c807255d2f9b8cd16860.png)
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2022-03-05更新
|
1915次组卷
|
3卷引用:重庆市第八中学校2022-2023学年高二下学期第二次月考数学试题
名校
解题方法
7 . 已知椭圆
的上、下顶点分别为A,B,离心率为
,椭圆C上的点与其右焦点F的最短距离为
.
(1)求椭圆C的标准方程;
(2)若直线
与椭圆C交于P,Q两点,直线PA与QB的斜率分别为
,
,且
,那么直线l是否过定点,若过定点,求出该定点坐标;否则,请说明理由.
![](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/c34e01955f8c8fe2f0041b35d8d602a7.png)
(1)求椭圆C的标准方程;
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fc5bd66dd6d5e09ff0893a938aed56e.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/ef42faa9d21371359b8a87584bc466f7.png)
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2022-03-05更新
|
315次组卷
|
2卷引用:河南省洛阳市2021-2022学年高二上学期期末考试数学(理)试题
名校
解题方法
8 . 已知椭圆
的离心率为
,右焦点为F,右顶点为A,且
.
(1)求椭圆C的标准方程.
(2)若不过点A的直线l与椭圆C交于D,E两点,且
,判断直线l是否过定点,若过定点,求出定点坐标;若不过定点,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/2515775bb766ebaec1323d42938c4c8a.png)
(1)求椭圆C的标准方程.
(2)若不过点A的直线l与椭圆C交于D,E两点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a087becc73721a258db03e347d0134f.png)
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2022-03-04更新
|
554次组卷
|
3卷引用:江西省赣州市赣县第三中学2021-2022学年高二3月月考数学(文)试题
名校
解题方法
9 . 已知椭圆
上一点
到两焦点的距离之和为
.
(1)求椭圆C的方程;
(2)不经过点
的直线l与x轴垂直,与椭圆C交于A,B两点,若直线BQ与C的另一交点为D,问直线AD是否过定点?若过定点,请求出定点坐标;若不过定点,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74448318cf1342634b794412cc7c7d88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38387ba1cadfd3dfc4dea4ca9f613cea.png)
(1)求椭圆C的方程;
(2)不经过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/451fc6e4248b63e70595f23842f06c93.png)
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2022-03-03更新
|
229次组卷
|
3卷引用:河南省新乡市2021-2022学年高二上学期期末考试数学(文)试题
名校
解题方法
10 . 已知椭圆
,过焦点且垂直于长轴的弦长为1,且焦点与短轴两端点构成等边三角形.
(1)求椭圆的方程;
(2)过点
的直线
交椭圆于
,
两点,交直线
于点
,且
,
.求证:
为定值,并计算出该定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
(1)求椭圆的方程;
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/173ffa0c1821a4817d499a541644815c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.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/a089c207e39a24d0d82aa853ac2bbb8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dae581e0d174441859c7a93c82a877df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c20538fa2a49b422015e795cb7b2c5ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/febf7413b35cf2889fdb57a6b519087c.png)
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2022-02-12更新
|
532次组卷
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2卷引用:广东省深圳市龙岗区2021-2022学年高二上学期期末数学试题