解题方法
1 . 如图,
,
分别是双曲线
(
,
)的左、右焦点,且
,过
的直线
与双曲线的左、右两支分别交于点
,
.若
为等边三角形,则双曲线的方程为______
![](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/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/6b69f4f4c0d380494e2aaf221af71a97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.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/e5eb2485f90dbfd0dfd6e7d179a856f5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/24/8bee17e0-f87c-40eb-bbe2-b1b82df3647a.png?resizew=182)
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名校
解题方法
2 . 已知双曲线
,
,
分别为其左、右焦点,过
作直线
轴交双曲线
于
,
两点,将双曲线所在的平面沿
轴折成一个锐二面角,设其大小为
,翻折后
,
两点的对应点分别为
,
,记
,若
,则该双曲线
的离心率为( )
![](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/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/165a501b2e6a3acc46212e59a166c053.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.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/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953cec61ac602ce5eb59b7912352179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/962e19a5ac7c58bc03966a5a024819f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3915b9a670b32ed18707a5db8be3c5bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
A.![]() | B.![]() | C.3 | D.2 |
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解题方法
3 . 已知双曲线
满足条件:(1)焦点为
;(2)离心率为
,求得双曲线的方程为
.若去掉条件(2),另加一个条件求得的双曲线的方程仍然为
.则下列四个条件中,符合添加的条件可以为____________ (填序号)
①双曲线上的任意一点P都满足:
;
②双曲线的虚轴长为4;
③双曲线的一个顶点与抛物线
的焦点重合;
④双曲线的渐近线的方程为:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5ec7fa23be9cbe9a50607ea6bc8a4ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9563e601d49dc33c997269bc80345584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caa585b9257ed0798213a9ae9b87d291.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a6af0c85c1a59c2b77b6e550e1ef54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a6af0c85c1a59c2b77b6e550e1ef54.png)
①双曲线上的任意一点P都满足:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53cb222280a591953f0568b879618f41.png)
②双曲线的虚轴长为4;
③双曲线的一个顶点与抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0261104a7c308433a0c0508ff20ea29a.png)
④双曲线的渐近线的方程为:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c953cbb95dc2f9e91efac4a67e92144e.png)
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4 . 已知等轴双曲线
的右焦点为
,过右焦点F作斜率为正的直线l,直线l交双曲线的右支于P,Q两点,分别交两条渐近线于M,N两点,点M,P 在第一象限,O是原点.
(1)求直线l斜率的取值范围;
(2)设
的面积分别为
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ffe6000b292a0ce7d93601d946e44c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8119a06465ad8c7f83665f932254960d.png)
(1)求直线l斜率的取值范围;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c10e02ba69df7f7f2e0d5568f75b7bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54432232d93b8be9a07987345751b4ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1da715809444c2d1e48da7d20a46ce3f.png)
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5 . 在平面直角坐标系
中,已知
,
为双曲线
的左、右焦点,
,
为C的左、右顶点,C的离心率等于2,P为C左支上一点,若
平分
,直线
与
的斜率分别为
,
,且
,则
等于___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.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/2c5c2e64358e0ec7aa142c336d970306.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef49a3ca580a144cc65a609c167facc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de1387f71028cfc9c6743260bebfd907.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ac86e1c253297a377e14fb9a1689be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/800c5e266b4ad8462a46970f0a232d52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860add72d63d440b55f10a52f2d51ac5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
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6 . 如图,已知双曲线
的左,右焦点分别为
,过
的直线与双曲线的右支交于
两点.若
,且
,则
的值为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/28/5205e9db-198e-459f-86bb-1a1d03bb4ef8.png?resizew=232)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7628c46ca7d36e7a885e9906ad77c18c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b799737e84f54908b8700fbaf1f96a80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a61d572ecf27dc02fcbd588f24647b1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b09e2d46f94b9ca3caf3f8283619c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49cb4a2984e5f9c5631fb1eed6173200.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7954f1485af8aa3199a249713a580ced.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13c62e7c58606b2250c6d189d85a6ea8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/28/5205e9db-198e-459f-86bb-1a1d03bb4ef8.png?resizew=232)
A.3 | B.2 | C.![]() | D.![]() |
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解题方法
7 . 如图所示,已知双曲线
与抛物线
有相同的焦点F,它们在第一象限内的交点为M.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/24/b8290a61-b8aa-4371-97c1-321311317c86.png?resizew=153)
(1)写出抛物线
的焦点坐标和准线方程;
(2)若双曲线
的焦距为其实轴长的2倍,求点M到双曲线
两个焦点的距离之和.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/211b9e53e4677ae9e2b20d5f7ce0a4e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/318874a31871355eb87e878ae35b71ad.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/24/b8290a61-b8aa-4371-97c1-321311317c86.png?resizew=153)
(1)写出抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39a52e26180930ad5b56a8a45f28a0f2.png)
(2)若双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/211b9e53e4677ae9e2b20d5f7ce0a4e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/211b9e53e4677ae9e2b20d5f7ce0a4e4.png)
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8 . 双曲线
的方程为
,左、右焦点分别为
,过点
作直线与双曲线
的右半支交于点
,
,使得
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce1132157a33c82610c2d5035493d024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01cb985b4dd5ea5648403c962261235f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b799737e84f54908b8700fbaf1f96a80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a61d572ecf27dc02fcbd588f24647b1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce1132157a33c82610c2d5035493d024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0914b68f106a912420705b2f3928ca42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1da9dcf6c319174c9ea2b1ceaed1649a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d04b221500ecec63d60773414147ea0d.png)
A.![]() | B.点![]() ![]() |
C.直线![]() ![]() ![]() | D.![]() ![]() |
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解题方法
9 . 已知
,则关于函数
说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5780ea14b5cce76d74b5bf57014d94df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09b29a7faa14a6e09d0db2d04f4ced03.png)
A.函数![]() ![]() | B.函数![]() ![]() |
C.![]() ![]() | D.![]() |
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解题方法
10 . 已知
分别是双曲线
的左、右焦点, 过
的直线与双曲线
的右支相交于
、
两点, 且
. 若
, 则双曲线
的离心率为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c5815cf137ffaeae73bd55c9a66b4a0.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9791f9921b0e33237809e8839992a8eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683b5dfdb789d67eb158c862b8b58f50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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