抛物线的性质

定理1　抛物线y2=2px(p>0)的焦点是F(p2,0).如果P(x0,y0)是抛物线上一点,那么以FP为直径的圆与y轴相切.切点是Q(0,y02).并且直线PQ是抛物线的切线.(如图1).证明　易求以FP为直径的圆的方程为x2 y2-(x0 p2)x-y0y 12px0=0由方程组x2 y2-(x0 p2)x-y0y 12px0=0x=0消去x得关于y的一元二次方程y2-y0y 12px0=0(1)考虑到P(x0,y0)点在抛物线y2=2px上,显然方程(1)的判别式Δ=y20-2px0=0,所以,以FP为直径的圆与y轴相切.易求切点是Q(0,y02).由两点式方程,考虑P(x0,y0)点在抛物线上,可求得直线PQ的方程为y0y=p(x x0),此即一般教科书上抛物线的切线…     

有关椭圆焦点的一些性质

在二次曲线的研究中,有些问题用射影几何的方法比用平面几何方法处理更简单、自然,且条理更清楚.用射影几何的方法,将二次曲线中的椭圆放在拓广平面上,给出椭圆特别是有关椭圆焦点的许多有趣性质.     

抛物线的三个性质

本文给出抛物线的三个性质以及与它们相对应的抛物线的三种画法;性质Ⅰ　设抛物线ｙ２＝２ｐｘ（ｐ＞０）,焦点为　图１Ｆ（ｐ２,０）,Ｑ是抛物线上除顶点外的任意一点,直线ＱＳ与ｘ轴平行,过点Ｆ作∠ＳＱＦ的角平分线的垂线,垂足为Ｐ,则点Ｐ的轨迹是ｙ轴（除去顶点Ｏ）;（如图１）证明　设直线ＱＳ与抛物线的准线ｌ,ｙ轴分别相交于点Ｓ和Ｖ,ＦＳ与ｙ轴交于点Ｐ′;易知ＳＶ＝ＯＦ,则Ｒｔ△ＳＶＰ′≌Ｒｔ△ＦＯＰ′;∴　ＳＰ′＝Ｐ′Ｆ．故　Ｐ′为线段ＳＦ的中点;由抛物线的定义得ＳＱ＝ＱＦ,所以△ＳＱＦ是等腰三角形;∴…     

有关直角双曲线的一些性质

近年,在二次曲线上的研究中,发现直角双曲线可以由它的内接三角形的垂心生成,且用射影几何的方法比用平面几何方法处理更自然、条理更清楚.在此基础上,用射影几何的方法得到一些直角双曲线的性质,给出了直角双曲线的其它生成方法.     

抛物线的两个直角性质

本文介绍抛物线的两个直角性质,供读者参考.定理1经过抛物线y2=2px(p>0)的准线和对称轴的交点E作斜率为k的直线,与抛物线的一个交点是P,F是抛物线的焦点,若∠EPF=     

抛物线焦点弦的性质

抛物线焦点弦具有不少性质 ,均散见在各类书刊上 .本文将系统地归纳集中 ,以期对焦点弦的几条最主要的性质有一个更全面的、更深刻的了解 .从而进一步提高运用这些性质去解决相关问题的数学素质和应用能力 .( 1 )1　焦点弦 (通径 )的定义通过抛物线焦点的直线(不与抛物线对称轴平行 )被抛物线截得的线段 ,叫做抛物线的焦点弦 ,如图 (1 ) .线段 AB叫做抛物线 y2 =2 px(p >0 )的焦点弦 . (当AB垂直于抛物线的对称轴时 ,AB叫做抛物线的通径 ) .2　焦点弦的性质定理 1　抛物线焦点弦长等于 2 p(1 1k2 )或2 psin2 α并且以通径长为最小 ,最小…     

《圆锥曲线焦点弦的一个性质》一文的补充和推广

在文 [1 ]中给出如下结论 :定理 1　设AB ,CD是圆锥曲线过焦点F的两动弦 ,弦端点连线AC ,BD交于点M ,则M的轨迹是圆锥曲线的相应准线 .本文对文 ( 1)的证明做些补充并给出定理1的推广形式 .1　补充在文 [1]中给出的定理 1的证明 ,其实是仅证出点M一定在准线上 ,还应补证 :准线上任意一点M ,都存在过焦点的两条弦AB ,CD使AC ,BD的交点为M .补充如下 :设点M( ρ0 ,θ0 )是圆锥曲线E的准线l:ρcosθ=-p上任意一点 ,过点M做直线AC交E于A( ρ1 ,θ1 ) ,C( ρ2 ,θ2 ) ,延长AF ,CF分别交E于B( ρ1 ′,θ1 π) ,D( ρ2 ′,θ2 π)…     

抛物线的一个几何性质

下面的定理 ,给出了抛物线一个有趣的几何性质 .此性质的证法很多 ,本文仅介绍一种较简捷的证法 .引理　设过点 (t,o) (t∈ R)的一条直线与抛物线 y2 =2 px(p >0 )相交于 P(x1,y1)、Q(x2 ,y2 )两点 ,则 x1x2 =t2 ,y1y2 =- 2 pt.证明　依题意可设直线方程为 x =my t,代入 y2 =2 px,得 y2 - 2 pmy - 2 pt=0∴　 y1y2 =- 2 pt,x1x2 =y212 p.y222 p=(y1y2 ) 24 p2 =(- 2 pt) 24 p2 =t2定理　设 A是抛物线 y2 =2 px(p >0 )的轴上一点 (位于抛物线内部 ) ,B是 A关于 y轴的对称点 .(1 )若过 A点引直线与这抛物线相交于 P、Q两点 (图 1 ) ,则∠…     

圆锥曲线两个性质的推广

《数学通报》2 0 0 2年第 6期文 [1 ]给出了圆锥曲线的如下两个性质 :性质 1　若Ｆ是圆锥曲线的焦点 ,Ｅ是与焦点Ｆ相对应的准线Ｌ和圆锥曲线对称轴的交点 ,ＡＢ是过焦点Ｆ的弦 ,ＢＣ∥ＦＥ ,点Ｃ在Ｌ上 ,则直线ＡＣ平分线段ＥＦ .性质 2　若Ｆ是圆锥曲线的焦点 ,Ｅ是与焦点Ｆ相对应的准线Ｌ和圆锥曲线对称轴的交点 ,ＡＢ是过焦点Ｆ的弦 ,点Ｃ在Ｌ上 ,直线ＡＣ平分线段ＥＦ ,则ＢＣ∥ＦＥ .本文旨在将以上两个性质进行推广 ,即若将性质中的焦点Ｆ推广为圆锥曲线 (包括圆 )对称轴上的任意一定点 ,是可得如下若干结论 ,1　性质 1的推广定理 1…     

抛物线的阿基米德三角形的性质

抛物线的阿基米德三角形的性质朱兆和（江苏省灌云县中学２２２２００）文［１］介绍了抛物线的阿基米德三角形和阿基米德定理．本文介绍阿基米德三角形图１的几个性质．性质１Ｍ（ｘ０，ｙ０）是抛物线ｙ２＝２ｐｘ（ｐ＞０）内一定点，则底边过定点Ｍ的所有阿基米德三角...     

Tracing Robert's use of representations to solve counting problems: A 16-year case study

This study describes how Robert used his external representations (formulae, notations, sketches, models, and figures) to solve progressively more challenging counting tasks over a 16-year period. In his explorations of counting tasks, Robert discovered intricate connections between solutions to problems that looked different on the surface. Using video data from Robert's problem solving, analyses of his solutions are presented that shed light on how he built new ideas from existing ideas and how he modified external representations to make new mathematical discoveries and provide justifications for his solutions.     

Some new mean value theorems of Flett type

In this paper, we give some new mean value theorems, which are generalizations of Flett, Myers and Tong's theorems.     

Stability criteria for differential equations with variable time delays

Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay differential equations arising in population dynamics may serve as good teaching examples for these methods. The determination of linear stability properties for an ordinary differential equation with a varying time delay is carried out through discrete point analysis, either by seeking explicit solutions or leading to the consideration of a difference equation and the roots of a characteristic polynomial. Numerical simulations carried out using MATLAB Simulink are compared to the analytical solutions, and computation is also used to suggest extensions to some results.     

从射影观点看焦点

用射影几何知识讨论欧氏平面上二次曲线焦点与准线的性质.     

Some Computational Aspects of Geodesic Convex Sets in a Simple Polygon

In this article, we deal with some computational aspects of geodesic convex sets. Motzkin-type theorem, Radon-type theorem, and Helly-type theorem for geodesic convex sets are shown. In particular, given a finite collection of geodesic convex sets in a simple polygon and an “oracle,” which accepts as input three sets of the collection and which gives as its output an intersection point or reports its nonexistence; we present an algorithm for finding an intersection point of this collection.     

Tribonacci-like sequences and generalized Pascal's pyramids

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascal's pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinberg's triangle associated to a suitable generalized Pascal's pyramid. The results also extend similar properties of Fibonacci-like sequences.     

Some problems of A. Kroó on multiple Chebyshev polynomials

Three problems of A. Kroó on multiple Chebyshev polynomials are solved using the Borsuk–Ulam antipodal theorem.     

On the Asymptotic Expansion of Hadamard Products

We consider functions f and g which are holomoxphic on closed sectors in C where they admit an asymptotic representation at ∞ in the form of power series in z^-1. We give a simple geometrical condition under which the Hadamard product f*g of f and g porsesses again an asymp totic expansion at ∞. It turns out that the asymptotic expansion of f*g is essentially the formal Hsdamd product of the asymptotic expansions of f and g. Our result yields a slight generaliestion of a well known theorem of W. B. Ford.     

A variant of Hardy's and Miyachi's theorems for the Bessel–Struve transform

In this paper, we establish new versions of Hardy's and Miyachi's theorems for the Bessel–Struve transform.