1 . 已知双曲线
的一条渐近线方程为
,右焦点为
.
(1)求C的标准方程;
(2)过点F且相互垂直的两条直线
和
分别与C交于点A,B和点P,Q,记
的中点分别为M,N,求证:直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a2cfa22139b3e9c9a73500e1ba19f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2bdeeb6f5e38e3464c357d00839a6ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90ce47fde921058026708a4321a0e213.png)
(1)求C的标准方程;
(2)过点F且相互垂直的两条直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b487a48fb03928254b978f9245418515.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
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名校
解题方法
2 . 如图,已知
为双曲线
上一动点,过
作双曲线
的切线交
轴于点
,过点
作
于点
,则
的值是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/1/1ea17240-c95e-4253-9bba-74445b15ba92.png?resizew=160)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ef66f4832adc43902055a7e6d258037.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e9e177bc924f0662d0b8dfd57bdb463.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb1688bb9734c30c6b89a81ffca965f8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/1/1ea17240-c95e-4253-9bba-74445b15ba92.png?resizew=160)
A.![]() | B.![]() | C.![]() | D.不确定 |
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3 . 已知双曲线
经过点
,离心率为
,直线
过点
且与双曲线
交于两点
(异于点
).
(1)求证:直线
与直线
的斜率之积为定值.并求出该定值;
(2)过点
分别作直线
的垂线,垂足分别为
,记
的面积分别为
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/913f78382630e50543e5f7192cae3ed3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b671cdde6baf9ab577330696ca8ff121.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)
(1)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d454c82d9e52747563d47b68099249.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5671fb25040a712a49e8c8148d67d300.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd615b70f45cf2c9a47a2c812ffe1339.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3637753af5ce86be9c23a9beb6b5067.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/884d40a97fd767e95f34f3b91ab8d84c.png)
您最近一年使用:0次
2024高三下·广东·专题练习
解题方法
4 . 双曲线
,左、右顶点分别为
为坐标原点,如图,已知动直线
与双曲线
左、右两支分别交于
两点,与其两条渐近线分别交于
两点,则下列命题正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef644115c956ed62c3da8310c6f67ecd.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/fbbbc9c5353894f2c93c205c3ac04f03.png)
A.存在直线![]() ![]() ![]() ![]() |
B.![]() ![]() |
C.若直线![]() ![]() ![]() ![]() |
D.若直线![]() ![]() ![]() ![]() |
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解题方法
5 . 已知双曲线
的离心率为
,其顶点到双曲线C的一条渐近线的距离为
.
(1)求双曲线C的标准方程:
(2)设
,
,D为AB的中点,作AB的平行线l交双曲线C于不同两点P,Q,直线
和
分别与双曲线C交于M,N两点,求证:M,N,D三点共线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31f8f7e40ba386c0a9675896b52752d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9868f77d5ab5073b6145f1c6d272122e.png)
(1)求双曲线C的标准方程:
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8748dc55e2f45bc37fc4d84d7310f79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3326927e4b01e981a19109633141e06a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bf25e032b5599ac49383de06e776365.png)
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解题方法
6 . 已知
分别为双曲线
的左、右支上的点,
的右焦点为
为坐标原点.
(1)若
三点共线,且
的面积为
,求直线
的方程.
(2)若直线
与圆
相切,试判断
是否为定值.若是,求出该定值;若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9407dd040f0fc6216e18f9b28bc45d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09224287f31db61f4335560cdaf28ae3.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76e955a4c2ae8528a6103a58e5c6e050.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea1f0417d8269f01d8e0bc1a8756e2ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b10e8abf8690e4b129466ddb918bcc94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52b33328faae2d2d4921900e97424de5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e06f6f4a685dab387f1f8241b4e26e98.png)
您最近一年使用:0次
2024-03-07更新
|
588次组卷
|
2卷引用:广东省2023-2024学年高三下学期百日冲刺检测数学试题
7 . 如图,已知椭圆
,双曲线
是
的右顶点,过
作直线
分别交
和
于点
,过
作直线
分别交
和
于点
,设
的斜率分别为
.
(1)若直线
过椭圆
的右焦点,求
的值;
(2)若
,求四边形
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ad5c24ed80bb57cb2ac412b7b0d74b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa4d9576d6ea109df455920c6e4eb8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.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/098a3e7d1f1890863b7483a98b618119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.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/fbdb21011ea821b91d539cb763aac649.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/3/02955b36-04e8-46a3-a53b-943e9332e972.png?resizew=175)
(1)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b881044b5c73db6fcce110525741b02.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/257beb71337358f5ccc57219d9153666.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
您最近一年使用:0次
解题方法
8 . 已知
、
是双曲线
的左、右焦点,直线
经过双曲线的左焦点
,与双曲线左、右两支分别相交于
、
两点.
(1)求直线
斜率的取值范围;
(2)若
,求
的面积.
![](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/c477e5ade921ffa8377c4719319380ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.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)
(1)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83eb5d528de0ec4cc22b6e61479beae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
您最近一年使用:0次
名校
解题方法
9 . 已知点
为双曲线
上的动点.
(1)判断直线
与双曲线的公共点个数,并说明理由;
(2)(i)如果把(1)的结论推广到一般双曲线,你能得到什么相应的结论?请写出你的结论,不必证明;
(ii)将双曲线
的两条渐近线称为“退化的双曲线”,其方程为
,请利用该方程证明如下命题:若
为双曲线
上一点,直线
:
与
的两条渐近线分别交于点
,则
为线段
的中点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22962a2ad892cb6b14ab039a06e8cdc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ea74737939c0f94c91229a7098f36ec.png)
(1)判断直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e314fafd906934931d5cdfb4ecad2fca.png)
(2)(i)如果把(1)的结论推广到一般双曲线,你能得到什么相应的结论?请写出你的结论,不必证明;
(ii)将双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23e56464f202655f76b90e0f89e91be0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09cad114964b344e7c9b3903a21354e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/006d86d8f686ca9afaa3ef23a519728a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9f5f4ad0caac5a0fecb64f3908d2290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
您最近一年使用:0次
2024-03-04更新
|
1278次组卷
|
5卷引用:广东省汕头市2024届高三第一次模拟考试数学试题
广东省汕头市2024届高三第一次模拟考试数学试题(已下线)第26题 圆锥曲线压轴大题(1)(高三二轮每日一题)(已下线)大招4 圆锥曲线创新问题的速破策略江苏省盐城市五校联考2023-2024学年高二下学期第一次学情调研检测(3月)数学试题山东省烟台市牟平第一中学2023-2024学年高二下学期3月限时练(月考)数学试题
10 . 已知双曲线
是双曲线
的左顶点,直线
.
(1)设直线
过定点
,且交双曲线
于
两点,求证:直线
与
的斜率之积为定值;
(2)设直线
与双曲线
有唯一的公共点
.
(i)已知直线
与双曲线
的两条渐近线相交于两点
,求证:
;
(ii)过点
且与
垂直的直线分别交
轴、
轴于
两点,当点
运动时,求点
的轨迹方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aae0adfafc94233b75cf5cf3d792678.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edce7895b7c18ff6ada4f8975b414657.png)
(1)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f55d12701014cf53071093e8739d089b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
(2)设直线
![](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)
(i)已知直线
![](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/fbbbc9c5353894f2c93c205c3ac04f03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14da7a2daa1f7e1a5e3127a3060b35a3.png)
(ii)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.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/c38b9fcc365f60c8896d093edb46a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ffbca194ac467f226654cbd444a1813.png)
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