名校
解题方法
1 . 在英语中,实数是Real Quantity,一般取Real的前两个字母“Re”表示一个复数的实部;虚数是Imaginary Quantity,一般取Imaginary的前两个字母“Im”表示一个复数的虚部.如:
.已知复数
是方程
的解.
(1)若
,求证
;
(2)若
,复数
且满足
,在复平面内
对应的点为
,当
取得最大值时,求点
的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/485391d2213215f1b645d77cb12e1651.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8646eaa05bfde39d27813c301a076420.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07b0e1c4446de1a064b9c11e96b288d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5be05f2449cfe1fb871512e40199b151.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b34d58c444d285fe7ac7f6ca8d8bd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db95214c54cc0785b910bcf465159555.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39e23f7a3a11395e3b7fc4c41f9d71e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6447fb3ff7619b7c9e77e728bb14d18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
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解题方法
2 . 阅读下面问题的解法:
求复数
的模的取值范围.
解:
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/26/45f4c249-026c-4fd3-9492-1ffd76fc7cf9.png?resizew=181)
如图所示,设点A的坐标为
,点B的坐标为
,则
即为点A、B之间的距离
.
∵点B的轨迹为以O为圆心,半径为1的圆,∴
,因此复数
的模的取值范围是
.
试运用类似上面的解法解下列问题:求函数
的值域.
求复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3441a58ede2f3bd780d0207c8047127.png)
解:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffe803d3c95c4d89ada302952dfccae1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/26/45f4c249-026c-4fd3-9492-1ffd76fc7cf9.png?resizew=181)
如图所示,设点A的坐标为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43d2ff9ced79c22ff48de8357b806412.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7265338f61b72fef35a0746c8b429876.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce0584993cfa0801d603ffca041ecf3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05da40a2d7dab5d6a003906ca19d4749.png)
∵点B的轨迹为以O为圆心,半径为1的圆,∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/946a7572a73a40b3af7a1a80722cc446.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3441a58ede2f3bd780d0207c8047127.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0691c8ee320d0f36d7122d3e5ab0b7ee.png)
试运用类似上面的解法解下列问题:求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/792b0cf62b3db17fc6ea267e0267f001.png)
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