解题方法
1 . 已知以
为左、右焦点的双曲线
的一条渐近线为
.点
是双曲线
上异于顶点的动点,若
是
的平分线上的一点,且
,则
的取值范围是_____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f0b0da26aac472886b8345cfb9bf8ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](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/c94fe48bf7af022ecbbe13833fdcc2c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff8398a001fcf2bc80b13cce4ca1caa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4586226c7e189c5f55f6e9f61c474efb.png)
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解题方法
2 . 已知双曲线
,点
为双曲线右支上的一个动点,过点
分别作两条渐近线的垂线,垂足分别为
,
两点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18465f3f5d9fcd336a25610fa93a34e8.png)
![](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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
A.双曲线的离心率为![]() |
B.存在点![]() ![]() |
C.四边形![]() ![]() |
D.四边形![]() ![]() |
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解题方法
3 . 已知双曲线
的一条渐近线方程为
,
为坐标原点,点
在双曲线
上.
(1)求双曲线
的方程;
(2)若直线
与双曲线
交于
两点,且
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2bdeeb6f5e38e3464c357d00839a6ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ceb2f0b901bc82b7c8d8c12953fb2f9.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/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92401170a0eaa13334060d44a1929d78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac35b1e8a952aac4f4cdaaf02d868d04.png)
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4 . 已知直线
与双曲线
有唯一公共点
,过点
且与
垂直的直线分别交
轴、
轴于
两点,则当
运动时,点
到
两点距离之和的最小值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea6d37b4c6d03441fd2966ee1e8af41d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e1fa37c4c826b5dcfebe86ab6177906.png)
![](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/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/2e32888a46a26efd6930f1b1ed5ea9a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aee82283f06cedef32eb15b87964f5d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ac7ac2fd4338e4afbdc871d1d55519d.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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名校
解题方法
5 . 已知双曲线
的离心率为
,点
分别是双曲线的左右焦点,过
的直线交双曲线右支于P,A两点,点P在第一象限,当直线PA的斜率不存在时,
.
(2)线段
交圆
于点B,记
的面积分别为
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f3fa0b40fb0d9b8c62e37316ab3b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49f103bf5165f87bbd99090ee8cc50fe.png)
(2)线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ac86e1c253297a377e14fb9a1689be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aa46bcef998ac174ca0f564fc39b667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ed4138590ae2a43a383663f3cff223e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc4ea29a512c7979a09adcff888ed7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51da7a6689fcd857bdafe8b9aef58231.png)
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解题方法
6 . 设双曲线C:
(
,
)的右焦点为F,点O为坐标原点,过点F的直线
与C的右支相交于A,B两点.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/24/1e8e31f1-e40d-4c06-b771-0bac1138391c.png?resizew=147)
(1)当直线
与x轴垂直,且
两点的距离等于双曲线C的实轴长时,求双曲线C的离心率;
(2)若双曲线C的焦距为4,且
恒成立,求双曲线C的实轴长的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f3fa0b40fb0d9b8c62e37316ab3b04.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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/24/1e8e31f1-e40d-4c06-b771-0bac1138391c.png?resizew=147)
(1)当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
(2)若双曲线C的焦距为4,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c22b86f2cc5dc2f8053711f932a57f9.png)
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解题方法
7 . 已知实数
、
满足
,则
的取值范围是( )
![](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/474b5ec72ba456745109c1a76749af74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49d453550044ac822c41bad34c82f46b.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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名校
解题方法
8 . 若双曲线
的一个焦点是
,且离心率为2.
(1)求双曲线
的方程;
(2)已知点
,过焦点
的直线
与双曲线
的两支相交于A,B两点,求直线MA和MB的斜率之和的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63309dbc3612815f6dbdee23d9a10adc.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b0fc22034c9db29b23c4cf9faf5af6.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)
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解题方法
9 . 在平面直角坐标系中,
,动点
满足
,点
的轨迹记为曲线
.
(1)求
的方程.
(2)已知
,过点
的直线
(斜率存在且斜率不为0)与
交于
两点,直线
与
交于点
,若
为圆
上的动点,试问
是否存在最小值?若存在,求出最小值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8877dab240332cc0389ed7d4eed2679.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a7e8af0d17052cb0fbf8f8afafc4dd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)求
![](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/6fd0825e68122a65426840fbf07cf296.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/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.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/adaa6a78419f8944d7d7ef598d67314b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d44e8bc37ed03f44470762748a8f942a.png)
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10 . 双曲线
:
的左右焦点分别为
,
,两条渐近线分别为
,
,过坐标原点的直线与
的左右两支分别交于
,
两点,
为
上异于
,
的动点,下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a44342c2ee26a279265225982499b71.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/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.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)
A.若以![]() ![]() ![]() |
B.若![]() ![]() |
C.过点![]() ![]() ![]() ![]() ![]() ![]() |
D.设![]() ![]() ![]() ![]() ![]() |
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