小课题研究:培养学生创新能力的有效途径——以圆锥曲线光学性质的证明为例

• 作 者: 杜志国
• 机 构: 昆山开发区高级中学,
• 出 处: 中小学数学(高中版)
• 发表日期: 2015年Z1期
• 关键词: 光学性质 角平分线 到角公式 课题小组 高考试题 焦半径 镜面反射 反射镜面 反射光线 证法

一、问题的提出某杂志刊登了一篇题为《基于一道高考试题的数理探究及其推广》的文章,对于圆锥曲线的光学性质做出论证和推广.但文中对于光学性质的几何证明却存在错误,其内容摘录如下:结论2:椭圆C:x~2/a~2+y~2/b~2=1(a>b>0),P为C上除长轴端点外的任意一点,∠F_1PF_2的角平分线PM与过P点的椭圆的切线l垂直.

First, the issue put forward A magazine published an article entitled “based on a college entrance examination mathematical exploration and its promotion” article, the optical properties of the conic to make a demonstration and promotion.But the text for the optical properties of the geometric proof there Error, its content is excerpted as follows: Conclusion 2: Elliptic C: x ~ 2 / a ~ 2 + y ~ 2 / b ~ 2 = 1 (a> b> 0), P is C in addition to the long axis at any point , The angular bisector PM of ∠F_1PF_2 is perpendicular to the tangent line l of the ellipse over P point.