已知椭圆
的左、右焦点分别为
,点
是椭圆
上的点,离心率
.
(1)求椭圆
的方程;
(2)点
在椭圆
上,若点
与点
关于原点对称,连接
并延长与椭圆
的另一个交点为
,连接
,求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d652f51fbff6b854370f8a05e31cf2e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afdc0e11b47339776099587b1b0f2df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/075ba8c6fb5ef7288cd3fed425c8e69e.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5128eed1a97ba334115f456b8c8d181.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03e61e6283ecec51965aad41eeac53cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e28b79d420afa5b32428f6242a7b4c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3975bf0083e450fb3105002d9a63b5aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98013a5042685a1db94249e70c62c09a.png)
更新时间:2017-04-23 16:09:08
|
相似题推荐
【推荐1】如图,曲线
由两个椭圆
和椭圆
组成,当a、b、c成等比数列时,称曲线
为“猫眼曲线”,若猫眼曲线
过点
,且a、b、c的公比为
.
![](https://img.xkw.com/dksih/QBM/2020/12/29/2624460070461440/2627556471996416/STEM/36cb17ee-1f98-4495-8b03-483811e28c7d.png?resizew=322)
(1)求猫眼曲线
的方程;
(2)任作斜率为
且不过原点的直线与该曲线
相交,交椭圆
所得弦AB的中点为M,交椭圆
所得弦CD的中点为N,直线OM、直线ON的斜率分别为
、
,求证:
为与k无关的定值;
(3)设
、
为椭圆
上的两点,直线OP、直线
的斜率分别为
、
,且
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb7f0cd04b25e517333647b2580114ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f8ef77a389b15074e79e324baa70219.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/461b96f2cb34e5d76e4b9f97568025c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://img.xkw.com/dksih/QBM/2020/12/29/2624460070461440/2627556471996416/STEM/36cb17ee-1f98-4495-8b03-483811e28c7d.png?resizew=322)
(1)求猫眼曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
(2)任作斜率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bcb166b53a49e393871bcb14a528792.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9275bd8ce17fcc4a786510b008414ab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80065003fe6677e9a6b4f0dfb9d1db0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e4f28029c21c82a381a782e4b16fcbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af58317391aa288533c7feee92f52eb1.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d6f5adf13b4214666292dd64b947741.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8c774fedfa4c42cd3cb7c234da8baa0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb7f0cd04b25e517333647b2580114ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f0009063fe00277645aff1be6e32471.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/151e09f3a383d7811c619eef3a6398cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4202e13ac133ddf7aacdbbd70d23dae.png)
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【推荐2】已知椭圆
的两焦点
,
的坐标分别为
和
,点
在椭圆上.
(1)求椭圆
的标准方程.
(2)过坐标原点的直线交椭圆
于
,
两点,点
在第一象限,
轴,垂足为
,连接
并延长交
于点
.
①证明:
是直角三角形:
②求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6b0dadb875cccce870b69409a476606.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/d018f81e005f1909a938928258d3333e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57f2b1dfa46512f0abc75d26512ab707.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bec55f638559c413950ce41bbda9afb.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过坐标原点的直线交椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae95e96ce568efee50145f8d017353df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf3d566704b44ea4ef1f99c37bd46902.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59a0a6891259bf27be2280c6b6ba7430.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59a0a6891259bf27be2280c6b6ba7430.png)
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【推荐1】直线
称为椭圆
的“特征直线”,若椭圆的离心率
.
(1)求椭圆的“特征直线”方程;
(2)过椭圆C上一点
作圆
的切线,切点为
、
,直线
与椭圆的“特征直线”相交于点
、
,
为坐标原点,若
取值范围恰为
,求椭圆
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79d2eda2018c2c517f4172a36bfd0f20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5de85df85401e7e8da683ea4a784963c.png)
(1)求椭圆的“特征直线”方程;
(2)过椭圆C上一点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d0b232e71f45caadf858e3bac06f1e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef74c4299221a967507c6a179337581a.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/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f10ac864ad5d1eaa746f047a36a8887b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15180714bd2f42f1d32c39ce1a196449.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
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【推荐2】如图,在平面直角坐标系xOy中,已知椭圆C:
(a>b>0)的离心率为
,点(2,1)在椭圆C上.
![](https://img.xkw.com/dksih/QBM/2016/5/20/1572643402964992/1572643408732160/STEM/fbfb0888-31f1-4cf5-b0db-2913f6866e48.png?resizew=154)
(1)求椭圆C的方程;
(2)设直线l与圆O:x2+y2=2相切,与椭圆C相交于P,Q两点.
①若直线l过椭圆C的右焦点F,求△OPQ的面积;
②求证: OP⊥OQ.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://img.xkw.com/dksih/QBM/2016/5/20/1572643402964992/1572643408732160/STEM/fbfb0888-31f1-4cf5-b0db-2913f6866e48.png?resizew=154)
(1)求椭圆C的方程;
(2)设直线l与圆O:x2+y2=2相切,与椭圆C相交于P,Q两点.
①若直线l过椭圆C的右焦点F,求△OPQ的面积;
②求证: OP⊥OQ.
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【推荐1】已知椭圆
的左右焦点分别为
,点
在椭圆上,且
与
轴垂直.
(1)求椭圆的方程;
(2)过
作直线与椭圆交于
两点,求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/850324c95d750d52ab82218b1d38cfb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e86dbcf83cd5d3421b3eed7be7dab32d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)求椭圆的方程;
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da6b93dbe5272a5167ff4e2918bec864.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6d630091e362f48d6dbc73811e07d41.png)
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【推荐2】已知椭圆
的一个焦点与短轴的两端点组成一个正三角形的三个顶点,且椭圆经过点
.
(1)求椭圆
的方程;
(2)设直线
与椭圆
交于
,
两点,且以线段
为直径的圆过椭圆的右顶点
,求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e829105388311d42b94ca2c2654851de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8575889e326a115f411cab094aaea99.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.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/15c0dbe3c080c4c4636c64803e5c1f76.png)
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【推荐1】已知椭圆
,过动点
的直线l交x轴于点N,交C于点A、P(P在第一象限),且M是线段
的中点,过点P作x轴的垂线交C于另一点Q,延长
交C于点B.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/26/48828dd1-d5eb-4209-b80f-37564cc15d6d.png?resizew=160)
(1)求椭圆C的焦距和短轴长;
(2)设直线
的斜率为k,
的斜率为
,证明:
为定值;
(3)求直线
倾斜角最小时
的斜率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69daca955a565fa537347dd0d93783f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d8a5982a53874dd3e97d9af6d7942ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8ffe24cf9f327aeb241225ab15ab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db8305c4ffbf876642440c3d28e91e9f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/26/48828dd1-d5eb-4209-b80f-37564cc15d6d.png?resizew=160)
(1)求椭圆C的焦距和短轴长;
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db8305c4ffbf876642440c3d28e91e9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7e08d5c04f0431fb57b33a01717b599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4ded6dbecb802a8a296e767c4b41ea4.png)
(3)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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【推荐2】已知椭圆
的右焦点为F,椭圆
.
![](https://img.xkw.com/dksih/QBM/2022/4/28/2967973908578304/2969179699650560/STEM/edae2333f4f24b1ea933ca0d5f97c53b.png?resizew=182)
(1)求
的离心率;
(2)如图:直线
交椭圆
于A,D两点,交椭圆E于B,C两点.
①求证:
;
②若
,求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ff1455a4045eb93f482c0751840aea7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd3a3269dd7a4539c7eec25df816b60a.png)
![](https://img.xkw.com/dksih/QBM/2022/4/28/2967973908578304/2969179699650560/STEM/edae2333f4f24b1ea933ca0d5f97c53b.png?resizew=182)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)如图:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b44c7d3a8499af18da17231d6b898274.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/495021c7da1c77e6ed1d1dd30e0be7bc.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27a5569b9f84d91f16f1762ef1e6d0c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/004104bafb5f30338123d4ea2b7fedde.png)
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