1 . 已知双曲线
的虚轴长为4,C的一条渐近线与曲线
在
处的切线垂直,M,N为C上不同两点,且以MN为直径的圆经过坐标原点O,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2105999d86d0256b5fb90f814b305f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b9643da0c0fea4f099f9a9133d6076.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9444dc753b82bcc92bc9de494a9d7547.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e1c67633edc332efdbd4ceaf9e6244f.png)
A.![]() | B.4 | C.![]() | D.2 |
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解题方法
2 . 双曲线C:
的左、右焦点分别为
、
,过
且倾斜角为
的直线为
,过
且倾斜角为
的直线为
,已知
,
之间的距离为
.
(1)求C的方程;
(2)若过点
的直线l与C的左、右两支分别交于
两点(点
不在x轴上),判断是否存在实数k使得
.若存在,求出k的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/247c5b95745e280f039f8daa90ddecb6.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/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d387d228f512ada68fc79c9d5775b077.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d387d228f512ada68fc79c9d5775b077.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41322821ce31416fdac8dd6e0aa41c71.png)
(1)求C的方程;
(2)若过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad57e3727b7bbd795b05332fbf9649e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad57e3727b7bbd795b05332fbf9649e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffb71288fd1d0585d76040f828ea5595.png)
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3 . 已知双曲线
的左,右焦点分别为
,双曲线C的虚轴长为2,有一条渐近线方程为
.如图,点A是双曲线C上位于第一象限内的点,过点A作直线l与双曲线的右支交于另外一点B,连接
并延长交双曲线左支于点P,连接
与
,其中l垂直于
的平分线m,垂足为D.
(2)求证:直线m与直线
的斜率之积为定值;
(3)求
的最小值.
![](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/21d532ce76942846df88c6f66112e50f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c3d2cba96f6f03520c0b3f6e4da03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ac86e1c253297a377e14fb9a1689be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0739793f234f8e86adc6177801ae7295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94fe48bf7af022ecbbe13833fdcc2c8.png)
(2)求证:直线m与直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6f8d8af68c89038ba94704fba2defdd.png)
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2024-05-05更新
|
1003次组卷
|
3卷引用:广西南宁市第三中学2024届高三下学期校二模数学试题
名校
解题方法
4 . 双曲线
上一点
到左、右焦点的距离之差为6,
(1)求双曲线
的方程,
(2)已知
,过点
的直线
与
交于
(异于
)两点,直线
与
交于点
,试问点
到直线
的距离是否为定值?若是,求出该定值;若不是,请说明理由,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b9b445242927a1bf30cb82a763af275.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/2b3c9967968279271d8cf1f9444c0ac1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b569572c8d9bf05d78d3ab741e68bb2.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/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b66a5b7813e902306477f91f9f4084cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/279563c3c055777ce1aa369a2ef54aed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639c3d2ff5ee566fcc1b69c65712a661.png)
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2024-04-12更新
|
2187次组卷
|
8卷引用:广西南宁市2024届普通高中毕业班第二次适应性测试数学试题
5 . 已知双曲线的右焦点为
,渐近线方程为
.
(1)求C的方程;
(2)记C的左顶点为A,直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe14b5d65ffd11fb9750cdbeec6e073e.png)
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名校
解题方法
6 . 已知双曲线
(
,
)的左顶点为
,过点
的动直线l交C于P,Q两点(均不与A重合),当l与x轴垂直时,
.
(1)求C的方程;
(2)若直线AP和AQ分别与直线
交于点M和N,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c5c2e64358e0ec7aa142c336d970306.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd99c5000629d7f49499d666e68f40d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8f3765334f8eb07d789a55915293c5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99ffac516145b99309bc1b248fc625ae.png)
(1)求C的方程;
(2)若直线AP和AQ分别与直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a089c207e39a24d0d82aa853ac2bbb8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61f44f5daa2dd6f6c5014ff1cc6749d9.png)
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2024-02-21更新
|
351次组卷
|
2卷引用:广西柳州高级中学2024届高三下学期5月适应性演练数学试卷
解题方法
7 . 已知双曲线
经过点
,一条渐近线的倾斜角为
.
(1)求双曲线
的方程;
(2)若点
,过双曲线的右焦点
的直线
交双曲线于
.以
为直径的圆是否恒过点
,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c451e1159edb958b5567ecf6fa23dae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19dda128fd8bd88de4cdd76739788db7.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/20ebaa32f4f1f4f807ca9aeb7fb29951.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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解题方法
8 . 在平面直角坐标系
中,已知双曲线
的右焦点为
,一条渐近线的倾斜角为
,点
在双曲线
上.
(1)求双曲线
的标准方程;
(2)若点
在直线
上,点
在双曲线
上,且焦点
在以线段
为直径的圆上,分别记直线
的斜率为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a2cfa22139b3e9c9a73500e1ba19f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037fb348109dc2063a268b10eb925a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73a855587b8ce641f20294992e27d420.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d599059e6b2c918ab15ee22611b6962.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ff6ae1209d4f0b0013a2299d211e6ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4757181824e15e0f21e5bdd55448783.png)
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2024-01-25更新
|
161次组卷
|
4卷引用:广西壮族自治区北海市2023-2024学年高二上学期期末教学质量检测数学试题
9 . 已知双曲线
的左、右焦点分别为
,双曲线C的右顶点A在圆
上,且
.
(1)求双曲线C的标准方程;
(2)若过点
的直线l交双曲线C的右支于E,F两点,Q为x轴上一点,满足
;试问
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/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/560adea7b0d4fbe4131fc41f3fcbd871.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd32b90b7f4918d1dcdb513a94e2f2e3.png)
(1)求双曲线C的标准方程;
(2)若过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff2e866eefdb38ad0b3a52ec90d0329b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a9ef6c1350d532984159e7870a1b132.png)
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名校
解题方法
10 . 已知双曲线
的右焦点为
,过
且与
轴垂直的弦长为12.
(1)求双曲线
的标准方程;
(2)过
作直线
与双曲线交于
两点,问在
轴上是否存在点
,使
为定值,若存在,请求出
点坐标,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ef66f4832adc43902055a7e6d258037.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14436636ec6a7aec09cb63cecf6e970d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)过
![](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/e52586ca2a3b783bc8092415e2d4bf6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d871b0b46194d7300950e04f085533d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
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