名校
解题方法
1 . 已知双曲线
,其中离心率为
,且过点
,求
(1)双曲线
的标准方程;
(2)若直线
与双曲线
交于不同的两点
,
,且
,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a2cfa22139b3e9c9a73500e1ba19f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b848246c11ebef783e4e50f35282774.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d09b0f43e2ab01bece17d3624d6ef9e9.png)
(1)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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c0b06dc01c30d13f64be2ac6a1d811e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adb36455e34352b972eb1fc005100daa.png)
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解题方法
2 . 已知抛物线
,
为其焦点,过
的直线与抛物线
交于
两点,
为
中点,过
两点分别作准线的垂线交准线于
两点,直线倾斜角为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4151d4a6aa73f2aff7f46f3f83256cea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
A.若![]() ![]() |
B.![]() |
C.![]() ![]() |
D.过![]() ![]() ![]() |
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3 . 已知抛物线
的焦点为
,准线为
,点
在抛物线
上,过
作
的垂线,垂足为
,若
(
为坐标原点),则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bffd29b2e8192d673308bdf8d5cb6cab.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82ea1be9b9b6bb12afa7e1ce703d1603.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/bfa2c333eb3a718785c053a900c10d9a.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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e430de9d088494fae303e65baf93a97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bffd29b2e8192d673308bdf8d5cb6cab.png)
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4 . 如图,这是一个落地青花瓷,其中底座和瓶口的直径相等,其外形被称为单叶双曲面,可以看成是双曲线
的一部分绕其虚轴所在直线旋转所形成的曲面.若该花瓶横截面圆的最小直径为
,最大直径为
,双曲线的离心率为
,则该花瓶的高为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e57171806f407a98dd8a796d4d2d6bbd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9871f2a312aaf3a19b40e4fb1a7693b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35361e76a7c85d1886728c8d0200b234.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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5 . 已知抛物线C关于x轴对称,且焦点在直线
上,则抛物线
的标准方程为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b71d062eb189dbebac9b3b1b4b35fd9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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6 . 如图,在多面体ABCDEF中,平面
平面ABCD,
是边长为2的等边三角形,四边形ABCD是菱形,且
,
,
.
平面ACF;
(2)在线段AE上是否存在点M,使平面MAD与平面MBC夹角的余弦值为
.若存在,请说明点M的位置;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d28c625d7ac6878957facc8274d459c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f945a69cf7e8213e50622125cde652f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b25eff69a4a0dc7a7ab183843303d333.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed6034301fc4110da89bdb0f46ad82ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5928c98b341b16d4b5a5b931d2929d.png)
(2)在线段AE上是否存在点M,使平面MAD与平面MBC夹角的余弦值为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dee14db57f0c762aad845cf5b4a243c0.png)
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7 . 某数学兴趣小组研究曲线
和曲线
的性质,下面同学提出的结论正确的有( )
甲:曲线
都关于直线
对称
乙:曲线
在第一象限的点都在椭圆
内
丙:曲线
上的点到原点的最大距离为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73d188d87a33706615d669cb0d220710.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac7be9aa01e9ac4ad0087b2942261a09.png)
甲:曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21e9feabc99f62ee569b460e61526e2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
乙:曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bced7457d2e2c88d336ce5e4534a148.png)
丙:曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
A.3个 | B.2个 | C.1个 | D.0个 |
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名校
解题方法
8 . 已知双曲线C:
的焦距为
,点
在C的渐近线上,则双曲线C的方程为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3040b6c904477030ecf8ba20b2b18759.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b91d650c2fc1a741fabdb333b09aeb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/530e5817131adf2c05b99ff18eb9060f.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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解题方法
9 . 已知椭圆
,
为其左右两个焦点,过
的直线与椭圆交于
两点,则
的周长为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b3cfb07fcf2c1bf055d517631d80b7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5eb2485f90dbfd0dfd6e7d179a856f5.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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10 . 如图,长方体
的底面
为正方形,
为
上一点.
(1)证明:
;
(2)若
平面
,求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/419aee8a92d4b6ec81bf250c9ddb12d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22adbc0da438220f9cace11b629d799b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/1/56634a11-d9cf-4f5e-81e7-5d74b1c1a8f1.png?resizew=114)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42b6c68ad9b2e22725f3cbf7c1a3f8dc.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5adf679c5b5063388202ee10d28ee8c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c66d99a6a8415ddad22bbed33b64cfb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4eb7e9ad5486cf1c5e506b20c5469e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c66d99a6a8415ddad22bbed33b64cfb.png)
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