用圆锥曲线的光学性质来探究圆锥曲线对定点张直角的弦问题

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文[1]、文[2]探究了圆锥曲线对定点张直角的弦问题,得出了直角弦所在直线的包络仍然是圆锥曲线的结论.我们对这个结论有以下思考:由于每一条直角弦所在直线都是包络图形(圆锥曲线)的切线,而圆锥曲线的光学性质正是圆锥曲线的焦点、切点与准线之间的关系,那么这类问题运用圆锥曲线的光学性质来解答是否更接近问题本质呢?我们基于此思路进行了以下探究.椭圆的光学性质椭圆L的长半轴为a,两 The paper [1], [2] explored the problem of conic curves for fixed-point straight-angle chords, and obtained the conclusion that the straight-line envelope of a straight-angled string is still a conic curve. We have the following thinking about this conclusion: since each right angle The straight line of the string is the tangent of the envelope pattern (conic curve), and the optical properties of the conic curve are the focal point of the conic curve, the relationship between the cut point and the quasi-line, then this type of problem is solved using the optical properties of the conic curve. Is it closer to the nature of the problem? We have conducted the following exploration based on this idea. The optical properties of the ellipse The long axis of the ellipse L is a, two
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