1 . 若双曲线
的一条渐近线为
,右焦点
到直线
的距离为
.
(1)求双曲线
的方程;
(2)设过焦点
的直线
与双曲线
的右支相交于
,
两点(不重合),
(i)求直线
的倾斜角的取值范围;
(ii)在
轴上是否存在定点
,使得直线
和
的斜率之积为常数,若存在,求出
的坐标,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45cc81cfaccc00aa4b7139de5a35a102.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fa5d6092f598c7da4796f965e40525a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4b8503f4706b8321e4e79a87eadea84.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设过焦点
![](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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(i)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(ii)在
![](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/b66a5b7813e902306477f91f9f4084cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e5c62f22d7afc5627fcb86599faa8e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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2 . 已知圆
,圆
,动圆
与这两个圆中的一个内切,另一个外切.
(1)求动圆圆心
的轨迹方程.
(2)若动圆圆心
的轨迹为曲线
,
,斜率不为0的直线
与曲线
交于不同于
的
,
两点,
,垂足为点
,若以
为直径的圆经过点
,试问是否存在定点
,使
为定值?若存在,求出该定值及
的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db0b84389245562cb66fff69ecc8cabc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/face7de26202d8c61b02a9420d0d6f98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)求动圆圆心
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若动圆圆心
![](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/269a51e0f77f63bae2df3dc8b1d4f455.png)
![](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/8455657dde27aabe6adb7b188e031c11.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/32c38dfd14dde969702dff97ef2270f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdcf431fad1fd535fc09b3a9895d89d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
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7卷引用:河南省创新发展联盟2023-2024学年高二上学期第四次联考(12月)数学试题
3 . 已知点
在双曲线
上.
(1)已知点
为双曲线右支上除右顶点外的任意点,证明:点
到
的两条渐近线的距离之积为定值:
(2)已知点
,过点
作斜率为
的动直线
与双曲线右支交于不同的两点
,在线段
上取异于点
的点
,满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ad01b0639b0b618c9128df2a5d1315c.png)
(i)求斜率
的取值范围:
(ii)证明:点
恒在一条定直线上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3654254401fc902c3cb4912969f21f88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2702066c515f9b77353cfba5f9e33c0.png)
(1)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f85f1a0c955c915aefe3fcdc9d7eed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.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/86e203b7c9a6600e0272c58a23733490.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86e203b7c9a6600e0272c58a23733490.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ad01b0639b0b618c9128df2a5d1315c.png)
(i)求斜率
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(ii)证明:点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
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名校
解题方法
4 . 如图,在平面直角坐标系
中,双曲线
的上下焦点分别为
.已知点
和
都在双曲线上,其中
为双曲线的离心率.
(2)设
是双曲线上位于
轴右方的两点,且直线
与直线
平行,
与
交于点
.
(I)若
,求直线
的斜率;
(II)求证:
是定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7795aec93c2c7ac2fd93e6747ca6516c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdf3827864712be547ff16252f43baae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6d2b272b441a87d2f253fae0b19eefd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29f5b0f68545a8d59361331bcb6c2b06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/481c543b01d90183de82d82433bfa49e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cabea664e61863b3b3279dbce607924e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3656055f5256cd06e636ea96e9f89c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e86dbcf83cd5d3421b3eed7be7dab32d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbad65b3d744b70da2480eee1cdb587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(I)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc2fda21bd0a804fcd893a65cefaa701.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cabea664e61863b3b3279dbce607924e.png)
(II)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/762d2c5304d4b9a6867a359dbecb0956.png)
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2024-01-03更新
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4卷引用:江西省上饶市广丰贞白中学2023-2024学年高二上学期1月考试数学试题
5 . 已知曲线C上的任意一点到直线
的距离是它到点
的距离的
倍.
(1)求曲线C的方程;
(2)设
,
,过点
的直线l在y轴的右侧与曲线C相交于A,B两点,记直线AM,BN的斜率分别为
,
,求直线l的斜率k的取值范围以及
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02a873e20e420dc0904e8cc90eb230fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84b7ac31193004dcb7a71e8f657ad897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9868f77d5ab5073b6145f1c6d272122e.png)
(1)求曲线C的方程;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/accf35598ee054f1bf8b6584641d6d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4b2bb30137f92479d11827ee769f001.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/771494b7ef947e7f17e5e83a9fd83ac3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a6f62ec9c0e58b3f1c363158269b14d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0721e40e4c6929a24c54d2035c4014a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e085294c9036bf21db3cc7dc783606.png)
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2024-01-02更新
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3卷引用:河南省九师联盟洛阳强基联盟2023-2024学年高二上学期12月联考数学试题
河南省九师联盟洛阳强基联盟2023-2024学年高二上学期12月联考数学试题重庆市部分学校2023-2024学年高二上学期1月阶段测试数学试题(已下线)微考点6-6 圆锥曲线中斜率和积与韦达定理的应用
解题方法
6 . 在平面直角坐标系
中,已知动点
、
,点
是线段
的中点,且点
在反比例函数
的图象上,记动点
的轨迹为曲线
.
(1)求曲线
的方程;
(2)若曲线
与
轴交于
两点,点
是直线
上的动点,直线
分别与曲线
交于点
(异于点
).求证:直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fce7e9ea620220fc04ed1218f27123fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbee7e43b7285e078d74cc46c093e76e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba7204f43679af6935e494c59d40c6ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f42b2a9736c8943106472a7398d2892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)若曲线
![](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/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11b141148d19998c842aee2e5b1de63a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
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3卷引用:安徽省滁州市明光市第三中学2023-2024学年高二上学期11月月考数学
安徽省滁州市明光市第三中学2023-2024学年高二上学期11月月考数学(已下线)专题03 圆锥曲线题型全归纳(九大考点)-【寒假自学课】2024年高二数学寒假提升学与练(人教A版2019)江西省部分学校2023-2024学年高二下学期开学考试数学试题
7 . 在平面直角坐标系
中,动点P与定点
的距离和它到定直线
的距离的比是常数
,设动点P的轨迹为曲线C.
(1)求曲线C的方程;
(2)以原点O为端点作两条互相垂直的射线与曲线C分别交于点M,N.求证:
是定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdfe87e29d5bd01cb142e40828ad5f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1d0aa9412dd7caf42cc71520e282328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e483f50a009f2f66b269528e213756e6.png)
(1)求曲线C的方程;
(2)以原点O为端点作两条互相垂直的射线与曲线C分别交于点M,N.求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd908ea0fea10e9ae17b63da04845468.png)
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2023-12-31更新
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3卷引用:安徽省江淮名校2023-2024学年高二上学期12月阶段性联考数学试题
安徽省江淮名校2023-2024学年高二上学期12月阶段性联考数学试题河北省石家庄市赵县河北赵县中学、高邑县第一中学、无极中学2023-2024学年高二下学期4月月考考试检测数学试题(已下线)专题03 圆锥曲线题型全归纳(九大考点)-【寒假自学课】2024年高二数学寒假提升学与练(人教A版2019)
名校
解题方法
8 . 已知双曲线C:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f3fa0b40fb0d9b8c62e37316ab3b04.png)
过点
,右焦点F为
,左顶点为A
(1)求双曲线C的方程
(2)动直线
交双曲线C于M,N两点,求证:
的垂心在双曲线C上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f3fa0b40fb0d9b8c62e37316ab3b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/593a2990a546f41f504787e699140614.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/562ccfda509492b6f6c7ea2b3443e0d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc1ad2c2e04d161603f1be718b33ed44.png)
(1)求双曲线C的方程
(2)动直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61c18d772ebbbf0ed2259befe83701ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98013a5042685a1db94249e70c62c09a.png)
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2023-12-30更新
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2卷引用:辽宁省大连市第八中学2023-2024学年高二上学期12月月考数学试题
解题方法
9 . 设双曲线
的左、右焦点分别为
,且焦距为8,一条渐近线方程为
.
(1)求双曲线
的方程;
(2)已知
是直线
上一点,直线
交双曲线
于
两点,其中
在第一象限,
为坐标原点,过点
作直线
的平行线
,
与直线
交于点
,与
轴交于点
,证明:点
为线段
的中点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e04775db8aa7e8de50b3c1d83b8350ba.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad6ab8f625c9264bab65717a8c032c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/183b6a0cef4256c9696a5bca31053da5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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4卷引用:河北省邢台市质检联盟2023-2024学年高二上学期第四次月考(12月)数学试题
名校
解题方法
10 . 已知点
是双曲线
上任意一点.
(1)求证:点
到双曲线
的两条渐近线的距离的乘积是一个常数;
(2)已知点
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fa6b2d7c06d7b3bd15feefe023bb0ca.png)
(1)求证:点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e95e84f5c91c910aaafc5e74dbfbdf59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d063ec7f9dbeba72fabf4437f9400e07.png)
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2023-12-26更新
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6卷引用:河南省部分重点中学2023-2024学年高二上学期12月质量检测数学试题