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1、高中数学第二册（上）第八章省 扬 职 高华 伟Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Evaluation only. Created with A
2、spose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.尝试探究 形成概念想一想：把一根绳子对折，固定一端，用笔尖把绳子拉紧能画出什么图形？为什么？ 试一试：如果把绳子分开，固定两端，用笔尖拉紧绳子画出的图形是什么吗？ 思考：如果两定点为F1、F2，运动形成椭圆的动点为M，那么在此画图过程
3、中，什么量有变化？什么量没有变？你能参照圆的定义，给椭圆下个定义吗？Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.椭圆的定义l平面上到两个定点的距离的和（2a）等 于定长 的点的轨迹叫椭
4、圆 。F1F2M椭圆定义的文字表述：椭圆定义的符号表述：l定点F1、F2叫做椭圆的焦点。l两焦点之间的距离叫做焦距（2C）。 （大于|F1F2 |） Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty
5、 Ltd.椭圆方程推导的准备1建系设点2找出动点适合的条件3条件坐标化得方程4化简方程Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.椭圆方程推导F1F2Mxyo设M(x,y)是椭圆上任意
6、一点,椭圆的焦距为2c(c0),M与 F1、F2的距离和等于常数2a。则F1(c,o)、F2（c,0)由椭圆的定义可知：MF1+MF2=2a化简整理得：(a2-c2)x2+a2y2=a2(a2-c2) 由椭圆定义可知，2a2c,即ac, 所以a2-c20 令：a2-c2=b2,其中b0,代入上式，得：b2x2+a2y2=a2b2,两边同除以a2b2,得：Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides
7、 for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.椭圆的标准方程1它表示：1椭圆的焦点在x轴2焦点是F1（-C，0）、 F2（C，0）3C2= a2 - b2 F1F2M0xyEvaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .N
8、ET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.椭圆的标准方程2它表示：1椭圆的焦点在y轴2焦点是F1（0，-C）、 F2（0，C）3C2= a2 - b2 F1F2M0xyEvaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5
9、Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.焦点在焦点在x x轴轴轴轴上上焦点在y轴上C C2 2= a= a2 2- b - b2 2 C C2 2= a= a2 2- b - b2 2F1F2M0xyF1F2 M0xyEvaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.S
10、lides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.判定下列椭圆的焦点在？轴，并指明 a2、b2，写出焦点坐标答：在 X 轴。（-3，0）和（3，0 ）答：在 y 轴。（0，-5）和（0，5 ）答：在y 轴。（0，-1）和（0，1）判断椭圆标准方程的焦点在哪个轴上的 准则：焦点在分母大的那个轴上。Evaluation only.Evaluation only. Created with Aspose.Slides for
11、.NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.将下列方程化为标准方程，并判定焦点在 哪个轴上，写出焦点坐标在上述方程中，A、B、C满足什么 条件，就表示椭圆？答： A、B、C同号，且A不等于B。Evaluation only.Evaluation only. Created with Aspose.Slides
12、for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.写出适合下列条件的椭圆的标准方程1 a=4，b=1，焦点在 x 轴2 a=4，c=150.5，焦点在 y 轴上求一个椭圆的标准方程需求几个量？答：两个。a、b或a、c或b、c3两焦点的坐标是（0,4),(0,4),椭圆上一 点P到两焦点的距离和等于104两个焦
13、点的坐标是（-2，0）和（2，0）, 并且经过点（2.5，-1.5）Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.1 椭圆的标准方程有几个？答：两个。焦点分别在 x 轴、y 轴。 2给出
14、椭圆标准方程，怎样判断焦点在哪个轴上答：在分母大的那个轴上。 3什么时候表示椭圆？答：A、B、C同号，且A不等于B。 4求一个椭圆的标准方程需求几个量？答：两个。a、 b或a、c或b、c Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2
15、004-2011 Aspose Pty Ltd.作 业 ：1 95页 1 ，2、3 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.例 平面内有两个定点的距离是8，写出到这两个定点的距离的
16、和是10的点的轨迹方程。分析：1判断：和是常数；常数大于两 个定点之间的距离。故，点的轨迹是椭圆。2取过两个定点的直线做 x 轴，它的线段 垂直平分线做 y 轴，建立直角坐标系，从 而保证方程是标准方程。3根据已知求出a、c，再推出a、b写出椭圆的标准方程。Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.练习 ：1 已知三角形ABC的一边 BC 长为6，周 长为16，求顶点A的轨迹方程答：Ev
