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解题方法
1 . 数学中有许多形状优美、寓意美好的曲线,如星形线(如图),若让一个半径为
的圆在一个半径为
的圆内部,沿着圆的圆周滚动,小圆圆周上的任一点形成的轨迹即为星形线,其方程为
,给出下列四个结论,正确的是__________ .
![](https://img.xkw.com/dksih/QBM/2021/3/13/2677190890668032/2683996552208384/STEM/487c4da65463407f9dfd89bb0ae3a25a.png?resizew=151)
(1)星形线的参数方程为:
(
为参数);
(2)若
,则星形线及其内部包含
个整点;(即横、纵坐标均为整数的点)
(3)曲线
在星形线
的内部(包含边界);
(4)设星形线围成的面积为
,则
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b7081727a19eba3003a689f78872086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/309ad39777b3791c60c6c2782fd1e41f.png)
![](https://img.xkw.com/dksih/QBM/2021/3/13/2677190890668032/2683996552208384/STEM/487c4da65463407f9dfd89bb0ae3a25a.png?resizew=151)
(1)星形线的参数方程为:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ba98a3cbae70b69aaa45d034f358b9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab839d8569171afab5ed55c22013aa72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ec714b7d7a203ecbcef9232ee7d40da.png)
(3)曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eaceba4bfaac6c7a6dff561d92d9096a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bb228a835eea0494d4422a42e77dd69.png)
(4)设星形线围成的面积为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/759b91e7d21377cc9c598ce08e7ee17d.png)
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2021-03-23更新
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176次组卷
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2卷引用:宁夏石嘴山市第三中学2020-2021学年高二下学期期末考试数学(理)试题