抛物线集合性质资料讲解.ppt

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,抛物线的简单几何性质（一）,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,M,是抛物线,y,2,=,2,px,（,p,0）上一点，若点,M,的横坐标为,x,0,，则点,M,到焦点的距离是,x,0,+,2,p,O,y,x,F,M,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,焦半径及焦半径公式,抛物线上一点到焦点的距离,P(x,0,，y,0,)在y,2,=2px上，,P(x,0,，y,0,)在y,2,=-2px上,P(x,0,，y,0,)在x,2,=2py上,P(x,0,，y,0,)在x,2,=-2py上,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,1、抛物线的范围:y,2,=2px,y取全体实数,x,y,X,0,抛物线的几何性质：,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2、抛物线的对称性 y,2,=2px,关于,x,轴对称,没有对称中心，因此，抛物线又叫做无心圆锥曲线。而椭圆和双曲线又叫做有心圆锥曲线,X,Y,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,定义：抛物线与对称轴的交点，叫做抛物线的顶点,只有一个顶点,X,Y,3、抛物线的顶点 y,2,=2px,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,所有的抛物线的离心率都是 1,X,Y,4、抛物线的离心率 y,2,=2px,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,基本点：顶点，焦点,基本线：准线，对称轴,基本量：焦准距,p,（决定抛物线开口大小）,X,Y,5、抛物线的基本元素 y,2,=2px,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,例1：,已知抛物线y,2,=2x,(1)设点A的坐标为(，0)，求曲,线上与点A距离最近的点P的坐标及相应的|PA|的值；,(2)若上题中A(2,0)，则结果如何？,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,例2：,斜率为1的直线经过抛物线y,2,=4x 的焦点,与抛物线相交于两点A、B,求线段AB的长.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,6、焦点弦和通径,通径是焦点弦中最短的弦，,通径|AB|=2p,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,设AB是抛物线y,2,=2px(p0)过焦点F,的一条弦。设A(x,1,y,1,),B(x,2,y,2,),AB的,中点M(x,0,y,0,),过A,B,M分别向抛物线,的准线作垂线，垂足为A,1,B,1,M,1,则,y,F,A(x,1,y,1,),O,B(x,2,y,2,),M,A,1,B,1,M,1,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,A(x,1,y,1,),(1)|AB|x,1,+x,2,+p,(2)x,1,x,2,=,y,1,y,2,=-p,2,X,y,F,O,B(x,2,y,2,),M,A,1,B,1,M,1,y,2,=2px(p0),Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,y,F,A(x,1,y,1,),O,B(x,2,y,2,),M,A,1,B,1,M,1,(5)证明:以AB为直径的圆与准线相切,y,2,=2px(p0),AM,1,B=Rt,A,1,FB,1,=Rt,N,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,练习1：,已知抛物线方程为,y,2,=4,x,，直线,l,过定点P（,-,2，1），斜率为k.则k为何值时，直线,l,与抛物线,y,2,=4,x,只有一个公共点；有两个公共点；没有公共点呢。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,提出问题,过抛物线 的焦点的一条直线和抛物线相交，两交点的纵坐标为 ，,求证：.（焦点弦的其中,一条性质）,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,探究1,过焦点的直线具有上述性质，反之，若直线,AB,与抛物线 的两个交点,A,，,B,的纵坐标为 ，且 ，那么直线,AB,是否经过焦点,F,呢？,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,探究2,既然过抛物线焦点的直线与其相交，交点的纵坐标的乘积是一个定值，那么过抛物线对称轴上其他任意一定点，是否也有这个性质呢?,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,探究3,设抛物线 上两动点,，且满足,，问,AB,是否恒过某一定点？,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,探究4,设抛物线 上两动点,，且满足,，,求,AB,中点,P,的轨迹方程.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,探究5,设抛物线 上两动点,，,O,为坐标原点，,OA,OB,，则直线,AB,是否过定点？,求,AB,中点,P,的轨迹方程,.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,探究6,设抛物线 上两动点,，,M,为该抛物线上一定点，且,MA,MB,，则直线,AB,是否过定点？,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,探究7,若,M,为抛物线 上一个定点，,A,、,B,是抛物线上的两个动点，且,(,r,为非零常数,),，求证：直线,AB,过定点。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,将,“,探究,6,”,的,“,直线,MA,与直线,MB,的倾斜角之差为,90,0,”,变为,“,直线,MA,与直线,MB,的倾斜角之和为,90,0,”,，即 ，,r,=1,直线,AB,过定点.,将,“,探究,6,”,的,“,直线,MA,与直线,MB,的倾斜角之差为,90,0,”,变为,“,直线,MA,与直线,MB,的倾斜角之和为,180,0,”,，直线,AB,不过定点，但可得到：,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,探究8,若,M,为抛物线 上一个定点，,A,、,B,是抛物线上的两个动点，且直线,MA,与直线,MB,的倾斜角互补，求证：直线,AB,的斜率为定值。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,设计意图：,培养学生研究数学问题的一般思想方法，一是考虑原命题的逆命题是否成立；二是考虑能否把原命题进行一般推广；三是考虑从原命题条件中还能推出什么结论？四是考虑把原命题进行适当变式进行拓展。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,问题,(2004年北京卷理),过抛物线 上一定点,，作两条直线分别交抛物线于 .当,PA,与,PB,的斜率存在且倾斜角互补时，求 的值，并证明直线,AB,的斜,率为非零常数.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,变式1,过抛物线 上一定点 ，作两条直线分别交抛物线于 ，若直线,AB,的斜率为定值 ，证明直线,PA,与,PB,的倾斜角互补.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,设动直线,AB,：,y,=,-,x,+,b,与抛物线,相交于两点 ，问在直线,MN,：,x,=2,上能否找到一定点,P,（坐标与,b,的值无关），使得直线,PA,与,PB,的倾斜角互补？,变式2,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,变式3,如图，抛物线 ，,过点,P,(1,0),作斜率为,k,的直线,l,交抛物线于,A,、,B,两点，,A,关于,x,轴的对称点为,C,，直线,BC,交,x,轴于,Q,点，当,k,变化时，探究点,Q,是否为定点？,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,练习1：,如图，定长为3的线段AB的两端点在抛物线,y,2,=,x,上移动，设线段AB的中点为M，求点M到y轴的最短距离。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,练习2：,正三角形的一个顶点位于坐标原点，另外两个顶点在抛物线y,2,=2px(p0)上，求这个三角形的边长。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,变式：,已知在抛物线y=x,2,上三个点A、B、C组成一个等腰直角三角形，且顶点B是直角顶点，,(1)设直线BC的斜率为,k,，求顶点B的坐标；,(2)求等腰直角三角形的面积的最小值。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,抛物线的对称性问题,例.已知直线过原点，抛物线的顶点在原点，焦点在,x,轴的正半轴上，且点A（,-,1，0）和B（0，8）关于直线的对称点都在抛物线上，求直线和抛物线的方程。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,