中考试题与圆心距问题解析

由于两圆位置关系是初中几何的重要内容 ,而两圆圆心距的变化会引起两圆位置关系的变化 ,因此涉及圆心距的问题在中考命题中倍受青睐 .设两圆的半径分别为Ｒ ,ｒ,圆心距为ｄ ,那么 .<1>ｄ >Ｒ +ｒ 两圆外离 ;<2 >ｄ =Ｒ +ｒ 两圆外切 ;<3>Ｒ -ｒ<ｄ <Ｒ +ｒ(Ｒ≥ｒ) 两圆相交 ;<4 >ｄ =Ｒ -ｒ(Ｒ >ｒ) 两圆内切 ;<5>ｄ <Ｒ -ｒ(Ｒ >ｒ) 两圆内含 .下面对涉及圆心距的中考试题进行分类解析 .一、求圆心距例 1　 ( 2 0 0 1年辽宁省沈阳市中考题 )已知两圆半径分别是 2和 3,圆心距是ｄ,若两圆有公共点 ,则下面结论正确的是 (　 ) .Ａ .ｄ =1…     

新老教材习题一例浅探

在新旧教材中都有这样一道题目 :问题 1　求经过两圆 x2 y2 6 x - 4=0和 x2 - y2 6 y - 2 8=0的交点 ,并且圆心在直线 x - y - 4=0上的圆的方程 (老教材《平面解析几何》全一册 (必修 ) P70 ;新教材《数学》第二册 (上 ) (试验修订本 .必修 ) P82 ) .此题一般有两种解法 .简解 1　将两圆的方程联立解得x =- 1 ,y =3. 　或　 x =- 6 ,y =- 2 .知两圆的交点为 A( - 1 ,3) ,B( - 6 ,- 2 ) .而圆心在线段 AB的垂直平分线 x y 3=0上 ,由 x y 3=0 ,x - y - 4=0 .　 解得圆心坐标为C( 12 ,- 72 ) ,又圆的半径为 r =| AC| =12 1 7…     

极坐标系下圆的方程

我们知道 ,在直角坐标系中 ,圆有标准方程和一般方程 ,那么在极坐标系中 ,圆的标准方程和一般方程又是怎样的呢 ?1　极坐标系下的圆求圆心是Ｃ( ρ0 ,θ0 ) ,半径是ｒ的圆的极坐标方程 .设Ｍ ( ρ ,θ)是圆上任意一点 ,根据余弦定理得ｒ2 =ρ2 ρ20 - 2 ρ0 ρｃｏｓ(θ -θ0 ) ,即 ρ2 - 2 ρ0 ρｃｏｓ(θ -θ0 ) ρ20 -ｒ2 =0 ( 1)方程 ( 1)就是圆心是Ｃ( ρ0 ,θ0 ) ,半径是ｒ的圆的极坐标方程 .我们把它叫做极坐标系下圆的标准方程 .把圆的标准方程展开得 ρ2 - 2 ρ0 ｃｏｓθ0 ·ρｃｏｓθ -2 ρ0 ｓｉｎθ0 ·ρｓｉｎθ ρ20 …     

引入圆的切割线的奇思与妙用

由于圆心到圆的切割线的距离总是等于或小于圆的半径,于是得到一个不等关系.依此,我们将一类函数、方程中的问题转化为圆心到圆的切割线距离问题,从而使问题顺利得到解决.下面列举几例阐述,供读者参考.例1若点P(x,y)的坐标满足方程xcosα+ysinα-4=0,则当角α在实数范围内变化时,     

由2013年高考山东卷(理)第9题引发的探究

1题目及解法题目(2013山东理-9)过点(3,1)作圆(x-1)~2+y~2=1的两条切线,切点分别为A,B,则直线AB的方程为()A.2x+y-3=0 B.2x-y-3=0C.4x-y-3=0 D.4x+y-3=0此题考查圆的切点弦方程.试题短小精悍,难易适中,解法多样.为了方便说明,记点P(3,1),圆心C(1,0).思路1:如图1,欲求直线AB的方程,需求出点A,B的坐标,即两条切线与圆的公共点,因此,可以先求出两切线的方程,与圆的方程联立,通过解方程组求出点A,B的坐标,写出直线AB的方程.由于     

用伸缩变换动态解决椭圆问题

题目如图,已知一个圆的圆心为坐标原点,半径为2．从这个圆上任意一点P向x轴作垂线段PP′,求线段PP′中点M的轨迹．解设点M的坐标为(x,y,),点P的坐标为(x_0,y_0),则x=x_0,y=((y_0)/2)．因为P(x_0,y_0)在圆x~2     

巧定象限，妙解高考选择题

解高考数学选择题 ,当选择支中有明显的象限区分时 ,可确定出 (或排除掉 )有关点、向量所在坐标平面内的象限 ,就可得到快速简捷的解答 .下面举例说明 .1　图形 (图象 )问题例 1　 (2 0 0 1年全国高考题 )极坐标方程 ρ =2ｓｉｎ(θ π4)的图形是 (　　 )(Ａ)　　　　 (Ｂ)　　　　 (Ｃ)　　　　 (Ｄ)图 1　例 1图解　由选项知 ,主要区别在于圆心所在象限不同 .化为直角坐标方程 ,得ｘ2 ｙ2 =2ｘ 2 ｙ ,即圆心 (- Ｄ2 ,- Ｅ2 )为 (22 ,22 ) ,知在直角坐标系中 ,该圆的圆心在第一象限 ,结合选项排除 (Ａ) ,(Ｂ) ,(Ｄ) ,故选 (Ｃ) .例 2　 …     

用几何画板求与直线、定圆相切的圆心轨迹

先看下面的问题及解答 :图 1已知圆Ｃ :(ｘ -2 ) 2+ｙ2 =1 ,一动圆与ｙ轴相切 ,又与圆Ｃ外切 ,试求这动圆的圆心的轨迹方程 .解　如图 1 ,设动圆的圆心为Ｏ1(ｘ ,ｙ) ,有|Ｏ1Ｃ|=|Ｏ1Ｐ|+|ＰＣ|=|Ｏ1Ｐ|+1 ,即　 (2 -ｘ) 2 +ｙ2 =ｘ +1 .因此所求动圆的圆心轨迹方程为ｙ2 =6ｘ -3 .当定圆的半径变化时 ,比如半径分别为 2、3时 ,上述解法是否仍然正确呢 ?答案是否定的 .我们可以通过几何画板来观察分析 .具体作法如下 :显示坐标系 ,作一长度为 1的线段ＡＢ ,以Ｃ(2 ,0 )为圆心、ＡＢ为半径画圆 ,由上述解法可知与ｙ轴相切且与 (ｘ -2 ) 2 +ｙ…     

巧用点在曲线内

解析几何中涉及到动直线与二次曲线相交问题 ,若能利用点在曲线内部求解 ,常能使问题化繁为易 ,迎刃而解 .以下举几例说明 .例 1　已知圆Ｃ :ｘ2 + ｙ2 - 2ｘ - 4ｙ - 2 0=0 ,直线ｌ:( 2ｍ + 1 )ｘ + (ｍ + 1 ) ｙ - 7ｍ -4=0 ,求证 :无论ｍ取何实数 ,直线ｌ与圆Ｃ恒相交 .分析 :判断直线与圆的位置关系 ,通常运用判别式或比较圆心到直线的距离与圆半径的大小 .这样运算量往往很大 ,若能确定动直线所过的定点在圆内 ,就能解 .证明　圆Ｃ :(ｘ - 1 ) 2 + ( ｙ - 2 ) 2 =2 5,易求直线ｌ过定点Ｐ( 3,1 ) ,且 ( 3- 1 ) 2 + ( 1- 2 ) 2 =5<2 5.即…     

求代数式最大（最小）值的基本原理

在高三复习课中 ,学生对以下例 1的解法提出疑问 .　图 1　例 1图例 1　 (1997年全国高考理科第 (2 5)题 )设圆满足 :①截 ｙ轴所得弦长为 2 ,②被ｘ轴分成两段圆弧 ,其弧长的比为 3∶1,在满足①②的所有圆中 ,求圆心到直线ｌ:ｘ- 2 ｙ =0的距离最小的圆的方程 .解　如图 1,设圆心为Ｃ(ａ ,ｂ) ,半径为Ｒ ,由条件①可得ａ2 12 =Ｒ2 ,由条件②得∠ＡＣＢ =90° ,得Ｒ2 =2ｂ2 ,两式消去Ｒ ,得2ｂ2 -ａ2 =1(1)设圆心Ｃ到直线ｌ:ｘ - 2 ｙ =0的距离为ｄ ,则ｄ=|ａ - 2ｂ|12 2 2 =55|ａ - 2ｂ| (2 )问题变为求ｄ取到最小值时ａ ,ｂ的值 .将 (…     

Linear slices close to a Maskit slice

We consider linear slices of the space of Kleinian once-punctured torus groups; a linear slice is obtained by fixing the value of the trace of one of the generators. The linear slice for trace 2 is called the Maskit slice. We will show that if traces converge ‘horocyclically’ to 2 then associated linear slices converge to the Maskit slice, whereas if the traces converge ‘tangentially’ to 2 the linear slices converge to a proper subset of the Maskit slice. This result will be also rephrased in terms of complex Fenchel–Nielsen coordinates. In addition, we will show that there is a linear slice which is not locally connected.     

Reduction of continuous symmetries of chaotic flows by the method of slices

We study continuous symmetry reduction of dynamical systems by the method of slices (method of moving frames) and show that a ‘slice’ defined by minimizing the distance to a single generic ‘template’ intersects the group orbit of every point in the full state space. Global symmetry reduction by a single slice is, however, not natural for a chaotic/ turbulent flow; it is better to cover the reduced state space by a set of slices, one for each dynamically prominent unstable pattern. Judiciously chosen, such tessellation eliminates the singular traversals of the inflection hyperplane that comes along with each slice, an artifact of using the templates local group linearization globally. We compute the jump in the reduced state space induced by crossing the inflection hyperplane. As an illustration of the method, we reduce the SO (2) symmetry of the complex Lorenz equations.     

The structure of slices over minimal logic

In [1], we introduced a classification of extensions of Johansson’s minimal logic J by means of slices and proved the decidability of the classification. In this article, we find sufficiently simple necessary conditions for the maximality of logics in the slices formulated in terms of frames. This makes it possible to describe an efficient procedure for computing the slice number of any finitely axiomatizable logic over J. The maximal logics of the upper slices are written down explicitly.     

Hopf bifurcation and the centers on center manifold for a class of three-dimensional Circuit system

In this paper, Hopf bifurcation and center problem for a generic three-dimensional Chua's circuit system are studied. Applying the formal series method of computing singular point quantities to investigate the two cases of the generic circuit system, we find necessary conditions for the existence of centers on a local center manifold for the systems, then Darboux method is applied to show the sufficiency. Further, we determine the maximum number of limit cycles that can bifurcate from the corresponding equilibrium via Hopf bifurcation.     

Area-Preserving Normalizations for Centers of Planar Hamiltonian Systems

It is well known that a nondegenerate center of an analytic Hamiltonian planar system can be brought to normal form by means of an analytic canonical change of coordinates. This normal form, that we denote by CNF, does not depend on the coordinate transformation. In this paper we give an elementary proof of these facts and we show some interesting applications of the machinery that we develop in order to prove them. For instance, we describe the space of coordinate transformations that bring a Hamiltonian nondegenerate center to its CNF, and we prove that they are all canonical when the center is non-isochronous. We also show that two Hamiltonian systems with a nondegenerate center are canonically conjugated if and only if both centers have the same period function.     

最小最大二点集覆盖问题分析及改进算法设计

本文考虑二中心问题的扩展问题-最小最大二点集覆盖问题。给定两个平面点集P₁和P₂,分别包含m和n个点,求两个圆分别覆盖P₁和P₂,并且要求两圆半径与两圆圆心距三者中的最大值最小。本文主要贡献在于分析半径变化过程中两个点集中心包之间最近距离的变化关系,其中中心包是点集所具有的一个特殊几何结构,所得到的结果改进了Huang等人之前给出的结果,并且通过该结果设计相应算法,所得到的算法复杂性是目前最好的。     

Slices and Levels of Extensions of the Minimal Logic

We consider two classifications of extensions of Johansson’s minimal logic J. Logics and then calculi are divided into levels and slices with numbers from 0 to ω. We prove that the first classification is strongly decidable over J, i.e., from any finite list Rul of axiom schemes and inference rules, we can effectively compute the level number of the calculus (J + Rul). We prove the strong decidability of each slice with finite number: for each n and arbitrary finite Rul, we can effectively check whether the calculus (J + Rul) belongs to the nth slice.     

古塔的形变研究

利用投影多边形模型确定古塔各层的中心坐标,采用最小二乘法建立线性模型,借助三维高次曲线方程建立古塔的曲率模型和挠率模型。使用Matlab数值实验完成了对所有模型的求解,详细地分析了古塔的倾斜、弯曲和扭曲的变形趋势,为古塔的纠偏和保护工作提出了建议。     

Thermal and fluctuation effects of nonisothermal nucleation in the supercritical region of nucleating center sizes

By considering the possible deviations of the parameters of the quasistationary distribution of the nucleating centers of the condensed phase from their values in equilibrium with a vapor-gas medium we have obtained a form of expression of the Zel'dovich-Frenkel' balance equation that has made it possible to establish a hierarchy of time scales of the nucleation process, study the stage of thermal relaxation of the two-dimensional distribution of the nucleating centers, and construct a solution for that distribution at the end of the stage of thermal relaxation in the supercritical region of nucleating center sizes.St Petersburg University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 1, pp. 138–153, October, 1992.     

小血管血流速度分布的一种计算方法

本文所提出的计算方法,其基础是对血液流动微连续统模型作了一种边界条件的改进,设想了血管内壁面上血细胞速度可能不为零.对于由Eringen所提出的关于刚性圆管中稳态血液流动方程,假设了血管内壁面上血细胞的旋转速度,及血细胞旋转速度分布曲线在管轴处的斜率,导出了计算血管中速度分布曲线的方法,并将按此理论计算而得的曲线与Bugliarello和Hayden在实验中测得的分布曲线及由Turk,Sylvester和Ariman所提出的计算公式的结果相比较.