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定长线段在抛物线上滑动的中点轨迹与包络线

2014-03-18 11:43:27| 分类：几何学 | 标签： |举报 |字号大中小订阅

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（一）抛物线方程：

（1）解析方程： x²=2py

（2）参数方程：

x=p*tanθ

y=(p/2)tan²θ

（二）定长线段：

（1）定长：L

（2）倾斜角：θ

（三）中点轨迹方程：

x=p*tanθ

y=(p/2)tan²θ+(1/8)(L²/p)cos²θ

与抛物线围成的面积=(π/8)L²

（四）包络线方程：

x=p*tanθ+(1/8)(L²/p)sin(2θ)cos²θ

y=(p/2)tan²θ+(1/8)(L²/p)[2-cos(2θ)]cos²θ

与抛物线围成的面积=(π/4)L²+(π/8)L⁴/(8p)²

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