1 . 已知
和
为椭圆
上两点.
(1)求椭圆
的标准方程;
(2)若过点
且斜率为
的直线
交
于另一点
,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc039740839539ed44c9b2a77f2ba620.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff6f19b42df64bcc25192f8e81b9417.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.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/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a855335176fc36a15017f50a8561348.png)
您最近一年使用:0次
解题方法
2 . 在空间几何体
中,四边形
均为直角梯形,
,
.
,求直线
与平面
所成角的正弦值;
(2)如图2,设![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84450b3e28430072479ce38d273b115a.png)
(ⅰ)求证:平面
平面
;
(ⅱ)若二面角
的余弦值为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/986ba572d8373df48c996f8c8611498c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3b9460251cd9fcf7906c912a6a74c1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/115e3461ce95865ecf6d5bda10e24109.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/334df1bb0fcc75915955f4c355d1a6e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71c54178d7baff3270dde770f64f59af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00bab17635a999236e8d2e35017a208d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c0bfeadcf17b2a45896071f07a4a5a.png)
(2)如图2,设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84450b3e28430072479ce38d273b115a.png)
(ⅰ)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e51838e395dfc9d9ef597d9e01f46272.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/134ef0b1a2669a09f05bd4dc2496f706.png)
(ⅱ)若二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7267f2934c256fd74e58cb62d685bba0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4483fbccb86e51db927f5e7e08e0b044.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aefd06c239145a2b6ae87a955aa51414.png)
您最近一年使用:0次
3 . 如图的实线部分是江南某公园内的一个月亮门的正面外部轮廓,它由三部分构成:①水平地平线
;②位于地平线
与离地
高的水平线
之间的是长半轴长为
的同一个椭圆的左、右两侧的一部分;③水平线
以上是半径为
的半圆.
(2)某货运公司计划搬运一批大型包装箱通过此门,包装箱能否通过此门取决于其横截面的形状和大小,若包装箱的横截面分别为正方形或正三角形,搬运过程中要求包装箱保持水平状态(横截面与地面垂直,且有一边保持水平),为方便搬运,你会提前告诉货运公司哪些信息?为什么?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56045dacb47a5909c193ad70beb81806.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f71a41641aa0d0e45a3c03d3d2c1196b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0b4f2a143735077f598fbab4f1452ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f71a41641aa0d0e45a3c03d3d2c1196b.png)
(2)某货运公司计划搬运一批大型包装箱通过此门,包装箱能否通过此门取决于其横截面的形状和大小,若包装箱的横截面分别为正方形或正三角形,搬运过程中要求包装箱保持水平状态(横截面与地面垂直,且有一边保持水平),为方便搬运,你会提前告诉货运公司哪些信息?为什么?
您最近一年使用:0次
解题方法
4 . 已知点
为椭圆
上任一点,椭圆的短轴长为
,离心率为
.
(1)求椭圆
的标准方程;
(2)若点
是抛物线
的准线上的任意一点,以
为直径的圆过原点
,试判断
是否为定值? 若是,请求出这个定值; 若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/014b99f5c93a4ce8cd6251c12c1d1b37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4ea998b0d4284e05684a27fd1cca7ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3eeae1143162b4d564f4b77e3f77cd8.png)
您最近一年使用:0次
5 . 如图,在斜三棱柱
中,
分别是
的中点.
平面
;
(2)若
,且
,求直线
与平面
所成角
的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed1db08240a186b4b563c0e3e3dfd99d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9cfe81171fa15f47d81ca8bd4369234.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6748d9b9948485c5ba87ca8751c6e053.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e0feac3ad1bef70d1849e6abb91bb2.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63c336cb7a3eec081f1de84c1de782db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90b92c6f2f64bb2768f518cab73483b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15e58659e6ee4d93650e2edb6d6f7ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
您最近一年使用:0次
解题方法
6 . (1)请写出由拋物线的定义推导抛物线的标准方程
的过程;
(2)设直线
与抛物线
交于
两点,且
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49d3113c33d21b940f2fc864a79ca4b2.png)
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ab466aedd6e176088d8dee7bc3e3aaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49d3113c33d21b940f2fc864a79ca4b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52586ca2a3b783bc8092415e2d4bf6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b004f3c29ca828a163255c25323e7cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
您最近一年使用:0次
解题方法
7 . 如图所示,在棱长为2的正方体
中,
分别是棱
的中点,点
是棱
上的动点,
.
时,证明:直线
平面
;
(2)若二面角
的大小等于
,求
的值;
(3)记三棱锥
的体积为
,试将
表示为
的函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cf9b288c48c73463a2f214f02b6952a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d82d579a717399137b8c6d475d33cd4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22adbc0da438220f9cace11b629d799b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de085db8e8960a3c7b6df69c5b336628.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/500d68f2678989a5ce7431cfd51b019d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f5830646a912c3a916beac4f88c116b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7a447dc58e10adb7c8014071651e7c9.png)
(2)若二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e4c4a631729b25fa503b8c069bbbdfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270800ba975e87235e8802a631e5f3a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(3)记三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b03980f99fa0f339388e564466e8b94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
解题方法
8 . 2024年4月30日17时46分,神舟十七号返回舱成功着陆,返回舱是宇航员返回地球的座舱.返回舱的轴截面可近似看作是由半个椭圆 和一段圆弧 组成的“果圆”.如图,在平面直角坐标系中,某“果圆”中圆弧经过椭圆的一个焦点
和短轴的两个顶点
与
.
(2)直线
交该“果圆”于A、B两点,求弦AB的长度(精确到0.01).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad7345600383230ab4325bd6310813f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c0573e2af8a0dc8c6a1c0af067a324f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ee8c50793afd59e6ab4a2be5a877759.png)
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
您最近一年使用:0次
9 . 已知椭圆
的左、右顶点分别为
、
,且椭圆
经过点
.
(1)求
的值,并求经过点
且与圆
相切的直线方程;
(2)设
为椭圆
上的一个异于
、
的动点,直线
、
分别与直线
相交于
、
两点,求
的最小值:
(3)已知椭圆
上有不同的两点
、
,且直线
不与坐标轴垂直,设直线
、
的斜率分别为
、
,求证:“
”是“直线
经过定点
”的充要条件.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0f1e4548f62c6e3f124656c76ee64d0.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/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c16dcb66cd0d298f31f4f9c7e3a5fdcb.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f240cccaf24af8a796abb95cb42be52e.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.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/a6da2235c42867f9a79007c3fc83fec9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca3bd41676f6b69acac00a292fe134cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.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/a3d87446c9ef0230285d9b08127fce5c.png)
(3)已知椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.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/411461db15ee8086332c531e086c40c7.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/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66de27301ae08a4154ed37bb4a261b6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff32d26c8d44f5fb4813a19c1030a9f5.png)
您最近一年使用:0次
解题方法
10 . 某款平安锁边缘形状可以看作平面内一个椭圆的两段“弧”
和以椭圆左右焦点
为圆心的两个半圆
组成,曲线
和曲线
交于点
,
. 如图1所示建立平面直角坐标系
,曲线
所对应的方程为
,
,曲线
所对应的方程为
,
.
的值及曲线
所在椭圆的离心率
的值;
(2)现要在平安锁上找一个点
作为装饰孔,要求过点
且法向量为
的直线
与曲线
交于
两点(如图2所示),满足
,求实数
的值;
(3)商家要设计一个菱形凹陷以嵌入平安锁,要求该菱形的四边与平安锁椭圆段
和圆弧段
均相切(如图3所示),求该菱形的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c38928a92bc4b44ed3c9b89769f5372.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.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/256c6e6f44ba10b13c5c156dfa9a8e35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a9a1b99d2e67b15b3dc2a313c2280d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d80b861ba40387cb2bcd04945f5a371a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa638791abcfab75e5b6ed14e2bd4c86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c705c76c26208dad2ddb402de38ef4c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69192437b5c64df4a1595496fe3cfe07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
(2)现要在平安锁上找一个点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbc5fa556127662f4e4b02438bc68631.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9625823db815a2c5071250dfb7f2093b.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/e52586ca2a3b783bc8092415e2d4bf6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6158e04b97ecbba27d503617d7d8803d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(3)商家要设计一个菱形凹陷以嵌入平安锁,要求该菱形的四边与平安锁椭圆段
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