2024·北京·模拟预测
名校
解题方法
1 . 已知椭圆
的左顶点为
,两个焦点与短轴一个顶点构成等边三角形,过点
且与
轴不重合的直线
与椭圆交于
两点.
(1)求椭圆
的方程;
(2)若过点
且平行于
的直线交直线
于点
,求证:直线
恒过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2375a27ead9549550676d4e6a2b47243.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0f33f27e2c96f019bc9be1ac55e52f6.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/7789a500686c7a73770404ead6af0590.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a61d77911527508524874b212a0937d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3171b3d11c6f4619e189677345357508.png)
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解题方法
2 . 如图,平面四边形
中,
,
.若
是椭圆
和双曲线
的两个公共焦点,
是
与
的两个交点,则
与
的离心率之积为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f79863ffcfa63117ca6741b20a48e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bca7c0662dc9715b7cff113673763c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
A.![]() | B.![]() | C.2 | D.3 |
您最近一年使用:0次
2024-05-01更新
|
817次组卷
|
2卷引用:广东省茂名市高州中学2023-2024学年高二下学期期中考试数学试题
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3 . 已知曲线![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fdc384aa20e27b9289497e741e35554.png)
,下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fdc384aa20e27b9289497e741e35554.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af6383d7fbb4630f424a6ad10fdfe2f4.png)
A.曲线![]() |
B.当![]() ![]() |
C.若![]() ![]() ![]() |
D.若![]() ![]() |
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4 . 已知动圆
(
为圆心)过定点
,且与定直线
相切.
(1)求动圆圆心
的轨迹方程;
(2)设过点
且斜率为
的直线与(1)中的曲线交于
、
两点,求
;
(3)设点
是
轴上一定点,求
、
两点间距离的最小值
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bacfc149ede71417fa599c21b5a84cb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b556b1a9944719cf423e90f8df16c773.png)
(1)求动圆圆心
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)设过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47153fdd73c0661fa460130082e30929.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/29aa828f2bd9a5e63ee58dcaa9d0d336.png)
(3)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6214b125be1a9950a6c3d5999829d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.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/c7041b9af5b111b67a0eb98430ceaa17.png)
您最近一年使用:0次
2024-04-24更新
|
218次组卷
|
2卷引用:广东省广州市育才中学2023-2024学年高二下学期期中数学试题
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5 . 圆锥曲线光学性质(如图1所示):从椭圆的一个焦点发出的光线经椭圆形的反射面反射后将汇聚到另一个焦点处;从双曲线右焦点发出的光线经过双曲线镜面反射,其反射光线的反向延长线经过双曲线的左焦点. 如图2,一个光学装置由有公共焦点
,
的椭圆
与双曲线
构成,一光线从左焦点
发出,依次经过
与
的反射,又回到点
路线长为
;若将装置中的
去掉,则该光线从点
发出,经过
两次反射后又回到点
路线长为
.若
与
的离心率之比为
,则
( )
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3557809c066e68395b614535a7675e76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3557809c066e68395b614535a7675e76.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/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3557809c066e68395b614535a7675e76.png)
![](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/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3557809c066e68395b614535a7675e76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/705d0cd198f423d1a7d6b9a7999c18ee.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
6 . 已知
为坐标原点,抛物线
的焦点为
,过
的直线
与
交
,
两点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.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/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
A.![]() | B.若![]() ![]() ![]() |
C.若直线![]() ![]() | D.![]() |
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7 . 已知点
是圆
上一动点,点
,线段
的垂直平分线交线段
于点
.当点
运动时,设点
的轨迹为E.
(1)求点
的轨迹方程;
(2)已知过点
的直线
分别交E于
和
,且两直线的斜率之积为1,设
的中点分别为
,探究
轴上是否存在定点
,使得
,若存在,求出定点;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0f9dc1bbc5193d7c1e6a2bc1f459990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/996dadd8544543637bd2d32a1f79579e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e86dbcf83cd5d3421b3eed7be7dab32d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cabea664e61863b3b3279dbce607924e.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/ac047e91852b91af639feec23a9598b2.png)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)已知过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0235e0e72a28b90c4abe7b84f3dfa9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c4cd264c97c1f261229925cc5a6761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c4cd264c97c1f261229925cc5a6761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72b1a8ebfbce5db9fe9d0e2c8b9598c7.png)
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解题方法
8 . 对称轴都在坐标轴上的双曲线过点
,
,斜率为
的直线
过点
.
(1)求双曲线的标准方程;
(2)若直线
与双曲线有两个交点,求斜率
的取值范围;
(3)是否存在实数
使得直线
与双曲线交于A,B两点,且点P恰好为AB中点?为什么?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d3f986a4f053c576c8a58c7debc8829.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b55968d5f6b29626b1303e3cfe3132f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd064fc631163ed5e461887aa53cf197.png)
(1)求双曲线的标准方程;
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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解题方法
9 . 已知双曲线
的左焦点为
,方向向量为
的直线l过
与双曲线左,右两支分别交于
,
两点且
,则双曲线离心率为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07fe34d47ef23e8884a5c0404e54ef54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.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/5c2e97d6fb41ec5eccfd1fe8f72a92a6.png)
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10 . 如图曲线AB是以O为对称中心的椭圆的四分之一,A,B分别为长、短轴端点; 现只用圆规确定该椭圆的右焦点位置,步骤:以____ 为圆心以______ 为半径画弧与x轴的交点.(填序号即可)①点O;② 点A;③点B;④ OB长;⑤ OA长.
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