巧用圆锥曲线的定义求动点轨迹方程

在解析几何教学中 ,求动点的轨迹方程历来是教学重要专题之一 ,而椭圆曲线的两种定义又是研究圆锥曲线各种性质的基本出发点 ,如果在求动点的轨迹方程中充分利用圆锥曲线定义 ,常常会达到言简意明、异曲同工的效果 .下面就其运用作一些举例介绍 ,以飨读者 .1　运用第一定义求动点轨迹方程例 1　如图 1,已知椭圆 x2a2 + y2b2 =1(a >b >0 ) ,点P为其上一点 ,F1,F2 为椭圆的焦点 ,∠F1PF2的外角平分线为l,点F2 关于l的对称点为Q ,F2 Q交l于R ,当P在椭圆上运动时 ,求动点R的轨迹方程 .解　∵l为∠F1PF2 的外角平分线 ,且F2 ,Q两点关于l…     

95年高考第26题的推广

９５年高考第２６题的推广２５０００１山东省实验中学木生问题已知椭园，直线：Ｐ是ｌ上一点，射线ＯＰ交椭圆于点Ｒ，又点Ｑ在ＯＰ上且满足｜ＯＱ｜·｜ＯＰ｜＝｜ＯＲ｜２．当点Ｐ在ｌ上移动时，求点Ｑ的轨迹方程，并说明轨迹是什么曲线．对问题的条件进行推广将椭圆推...     

有心圆锥曲线的一个有趣性质

有心圆锥曲线的一个有趣性质李宗奇（甘肃省徽县一中７４２３００）１９９５年高考数学试题理科最后一题为：已知椭圆三十头一１，直线Ｚ：壬十兰一１．Ｐ是ｌ上一点，射线ＯＰ交椭圆于点Ｒ，又点Ｑ在ＯＰＩ且满足：当点Ｐ在ｌ上移动时，求点Ｑ的轨迹方程．并说明轨迹是什...     

如何确定离心率的范围

圆锥曲线的第二定义是:平面内动点M到定点F的距离和到一条定直线l的距离的比是常数e的轨迹是圆锥曲线．当01时,动点M的轨迹是双曲线,当e=1时,动点M的轨迹是抛物线．求椭圆与双曲线离心率的范围是高考的一类题型．下面从几个方面浅谈如何确定椭圆、双曲线离心率e的范围．     

平面内到两个定点的距离的和等于常数的点的轨迹一定是椭圆吗?

考察一道选择题.满足|z-1| |z 1|=1在复数Z在复平面内对应的点(A)轨迹是椭圆;(B)轨迹是双曲线;(C)轨迹是圆;(D)轨迹是一条线段;(E)轨迹不存在。 不少同学这样分析:根据复数的几何意义,方程|z-1| |z 1|表示动点到两个定点的距离之和等于常数。再根据椭圆的定义,该动点的轨迹是椭圆。故应选(A)。 其实,选(A)是错误的。 证明:(反证法)若(A)正确,那么椭圆的两焦点是F,     

一道课本习题的错解及剖析

高中数学第二册 (上 ) (试验修订本·必修 )P1 0 3上有这样一道习题 :点P与一定点F( 2 ,0 )的距离和它到一定直线x =8的距离的比是 1∶2 ,求点P的轨迹方程 ,并说明轨迹是什么图形 .常见解法 :由椭圆的第二定义及性质得 :c=2ca=12 a =4 b=2 3于是点P的轨迹是椭圆x21 6+y21 2 =1这种解法靠得住吗 ?不妨再看一例 :点P与一定点F( 1 ,0 )的距离和它到一定直线x =5的距离的比是 1∶ 3 ,求点P的轨迹方程 .错解 1 :同上例得所求的方程为x23 +y22 =1 .错解 2 :由椭圆的性质得c=1a2c=5 a2 =5,b2 =4.于是所求的方程为 x25+y24=1 .错解 3 :由椭圆的…     

由椭圆切线引出的一类有趣命题

《数学通讯》曾在2006年第15期上刊登了本人的拙作《椭圆切线的几个典型性质》,经进一步深入研究,笔者发现椭圆的切线在一定的条件下还可以引出一类有趣的命题——某些动点的轨迹仍然是椭圆．下面略举几个,并加以证明．     

定长弦的中点轨迹方程的探究

若弦长一定,当弦的两端点在曲线或面上运动变化时,其中点的轨迹图形是否存在?方程能否求出?这一问题很有趣,值得我们作一探究.本文将通过具体实例,对定长弦的两端点在直线、圆、椭圆、双曲线、抛物线及空间中的线或者面上运动时,弦的中点轨迹及其方程加以探究.一、求定长弦中点的轨迹     

一类解几问题的拓广

文[1]对下列问题进行了探讨,即 　　问题1设椭圆方程为x2/a2+y2/b2=1(a＞b＞0),椭圆内有一定点P(m,0),过该点作直线交椭圆A,B两点,又在该直线上另有一点Q,满足|AP/PB|〗=|AQ/QB|,求Q点的轨迹方程.……     

圆锥曲线统一定义的应用

椭圆、双曲线、抛物线统称为圆锥曲线.它们表示到定点F和定直线l的距离的比是一个常数e的点M的轨迹.当01时,点M的轨迹是双曲线;当e=1时,点M的轨迹是抛物线.其中定点F叫做焦点;定直线l叫做准线;定比e叫做离心率.这样的     

Minimum-Area ellipse containing a finite set of points. I

From the geometric point of view, we consider the problem of construction of a minimum-area ellipse containing a given convex polygon. For an arbitrary triangle, we obtain an equation for the boundary of the minimum-area ellipse in explicit form. For a quadrangle, the problem of construction of a minimumarea ellipse is connected with the solution of a cubic equation. For an arbitrary polygon, we prove that if the boundary of the minimum-area ellipse has exactly three common points with the polygon, then this ellipse is the minimum-area ellipse for the triangle obtained. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 980–988, July, 1998.     

Calculating ellipse area by the Monte Carlo method and analysing dice poker with Excel at high school

This paper reports on lessons in which 18–19 years old high school students modelled random processes with Excel. In the first lesson, 26 students formulated a hypothesis on the area of ellipse by using the analogy between the areas of circle, square and rectangle. They verified the hypothesis by the Monte Carlo method with a spreadsheet model developed in the lesson. In the second lesson, 27 students analysed the dice poker game. First, they calculated the probability of the hands by combinatorial formulae. Then, they verified the result with a spreadsheet model developed in the lesson. The students were given a questionnaire to find out if they found the lesson interesting and contributing to their mathematical and technological knowledge.     

椭圆积分的计算及其应用

椭圆积分的概念、椭圆积分表的编造以及通过实例说明椭圆积分的重要性.     

Investigation of boundary value problems for plane-parallel fluid flows in an anisotropic porous medium

We pose and consider the first and second boundary value problems and the transmission boundary value problem for plane-parallel steady flows in an anisotropic porous medium characterized by the permeability tensor, which is not necessarily symmetric. If the anisotropic medium is homogeneous, then the solutions of the problems in the case of canonical boundaries (a straight line or an ellipse) can be found in closed form, and in the case of arbitrary smooth boundaries, the study of these problems can be reduced with the use of Cauchy type integrals to the solution of inhomogeneous integral equations of the second kind. These problems are mathematical models of topical practical problems that arise, for example, in fluid (water or oil) recovery from natural soil strata of complicated geological structure.     

A computational study of the error in the vortex method associated with discontinuous vorticity

We investigate computationally the error computed by the vortex method for a discontinuous patch of vorticity. Specifically, the computed velocity and vorticity of an elliptical path of constant vorticity, known as the Kirchhoff ellipse, are compared to the analytic velocity and vorticity. The error in the velocity and the vorticity for the Kirchhoff ellipse as computed by the vortex method is presented. This error is studied as a function of the aspect ratio of the ellipse, the blob function, the spacing between the centers of the computational elements, and the blob radius. Both the error at the initial time and the error after three revolutions of the ellipse are discussed. © 1996 John Wiley & Sons, Inc.     

Minimum-Area ellipse containing a finite set of points. II

We continue the investigation of the problem of construction of a minimum-area ellipse for a given convex polygon (this problem is solved for a rectangle and a trapezoid). For an arbitrary polygon, we prove that, in the case where the boundary of the minimum-area ellipse has exactly four or five common points with the polygon, this ellipse is the minimum-area ellipse for the quadrangles and pentagons formed by these common points. Translated from Ukrainskii Matematicheskii Zhurnal, Vol.50, No.8, pp. 1098–1105, August, 1998.     

Probing interactions in exploratory teaching: a case study

This study investigates an exploratory teaching style used in an undergraduate geometry course to help students identify an ellipse. We attempt to probe beneath the surface of exploration to understand how the actions of teachers can contribute to developing students’ competence in justifying an ellipse. We analyse the complex interactions between student, content, and teacher, and discuss explicit pedagogical strategies that help students develop a higher level of geometric reasoning. The findings indicate that students engaged in guided explorations by the teacher and in group discussions with peers were able to identify an ellipse and justify their reasoning.     

On the boundary integral formulation of the plane theory of elasticity with applications (analytical aspects)

A formulation of the plane strain problem of the theory of elasticity in stresses, for simply connected domains, is carried out in terms of real functions within the frame of what is known as the boundary integral method. Special attention is devoted to the problem of determination of the arbitrary constants appearing in the solution, in view of work in progress where numerical techniques are used. Relying on some mathematical results formulated in the appendix, simple applications concerning the first and the second fundamental problems for the circle and for the ellipse are given, which show the correctness of the formulation and the necessity of recurring to numerical techniques, once the geometry of the problem or the type of boundary conditions deviates from being simple. Following parts of the present work are devoted to the numerical treatment of the obtained system of equations, as well as to the theories of thermoelasticity and thermo-electromagneto-elasticity.     

Double Aztec diamonds and the tacnode process

Discrete and continuous non-intersecting random processes have given rise to critical “infinite-dimensional diffusions”, like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary.     

On surface radiation conditions for an ellipse

We compare several On Surface Radiation Boundary Conditions in two dimensions, for solving the Helmholtz equation exterior to an ellipse. We also introduce a new boundary condition for an ellipse based on a modal expansion in Mathieu functions. We compare the OSRC to a finite difference method.