1 . 高中教材必修第二册选学内容中指出:设复数
对应复平面内的点
,设
,
,则任何一个复数
都可以表示成:
的形式,这种形式叫做复数三角形式,其中
是复数
的模,
称为复数
的辐角,若
,则
称为复数
的辐角主值,记为
.复数有以下三角形式的运算法则:若
,则:
,特别地,如果
,那么
,这个结论叫做棣莫弗定理.请运用上述知识和结论解答下面的问题:
(1)求复数
,
的模
和辐角主值
(用
表示);
(2)设
,
,若存在
满足
,那么这样的
有多少个?
(3)求和:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b40b6895776e0807c2baecbc8f33a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1116c1a2be36c2952f3f621854433824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/983fa8f4d178a0a909226523a33d521c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b40b6895776e0807c2baecbc8f33a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/437f03842c607c5554d86177ce090def.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eec3e684af41f9ed4db5b931b9ccfb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f56cfb41ee7cb758fee138ab09e0d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30cac4804764e9ffa2a2c9c37e450713.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6481f56ecdb2488e91835028d3cc7e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b604ddba45cd6dbf1b937f9db82906d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77476f0974841f574785fc9940b2f8ca.png)
(1)求复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/042b282f488b75517fb269e8b2512125.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e1d604600d084879cf3199cd0282345.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce48af55c99256efdc68fac0767d944c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f56cfb41ee7cb758fee138ab09e0d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f3b1a317184018ea9efc8154a878658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65a40142c84be68ee2918c3a8303388c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/388d3d213a231cccf854a29eef611d01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dffae22ae38d7238130e81a9e554d94b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)求和:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f152097ab61600de85e8181d056dab9b.png)
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2024-06-12更新
|
160次组卷
|
2卷引用:湖南省邵阳市第二中学2024届高三下学期5月模拟考试数学试题
解题方法
2 . 已知复平面上的点
对应的复数
满足
,设点
的运动轨迹为
.点
对应的数是0.
(1)证明
是一个双曲线并求其离心率
;
(2)设
的右焦点为
,其长半轴长为
,点
到直线
的距离为
(点
在
的右支上),证明:
;
(3)设
的两条渐近线分别为
,过
分别作
的平行线
分别交
于点
,则平行四边形
的面积是否是定值?若是,求该定值;若不是,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d9e97f066db90a8b341f8bc1cc3443d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e58bf28d4fde2909e1018e870e70baa9.png)
(1)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5dead5c5455fcbf21c809120dca4787.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9a2df68b4bc2f1773ccc4d4590079cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a9a1a2d4399061fc9d8921e22e1771e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51b892bab45e209077e2ac309bcc6428.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91edc7e2d4811f5ea6c01284cf00393a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c3529e7875112e656eed532629c5d3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c3529e7875112e656eed532629c5d3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/224150f5b61706dc52b162d76ee5e285.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f9f59db2b6f4b68d28271c9727afc0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4b5a39cff02408146d83d7704aa4d44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/449b78bc33d57b2972713b6029f39c32.png)
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