高中数学：圆的极坐标方程

1.3简单曲线的极坐标方程Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.教学目标1、认识几种圆的极坐标方程，比较它与直角坐标方程的异同。2、掌握求圆的极坐标方程的方法。3、能应用极坐标方程解决圆与直线的关系。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.教学重点：

求圆的极坐标方程的方法与步骤教学难点：极坐标方程是涉及长度与角度的问题，列方程实质是解直角或斜三角形问题，要使用旧的三角知识。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.答：与直角坐标系里的情况一样，求曲线的极坐标方程就是建系－设点（点与坐标的对应）－列式（方程与坐标的对应）－化简得方程f(,)=0

－说明求曲线的极坐标方程的步骤：Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.新课引入：热身训练：在平面直角坐标系中1、圆心坐标为(3,0)且半径为3的圆方程为

（x-3）2+y2=92、圆心坐标为（0,3）且半径为3的圆线方程为_______3、圆心在原点半径为3的圆方程为_______

X2+(y-3)2=9X2+y2=9变式：将以上三个方程化为极坐标方程并画出对应的图形。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.直角坐标系极坐标系极坐标图形1、圆心（3，0）半径为3圆心（3，0）半径为32、圆心（0，3）半径为3圆心（3,／2)3、圆心（0，0）半径为3圆心在极点，半径为3（x-3）2+y2=9X2+(y-3)2=9X2+y2=9

＝6cos

＝6sin

＝3xOC(3,／2

)xO3C(3,0)xOEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.曲线的极坐标方程定义：如果曲线Ｃ上的点与方程f(,)=0有如下关系(１)曲线Ｃ上任一点的坐标(所有坐标中至少有一个)符合方程f(,)=0；(２)方程f(,)=0的所有解为坐标的点都在曲线Ｃ上。

则曲线Ｃ的方程是f(,)=0。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.题组练习1求下列圆的极坐标方程(１)中心在极点，半径为r；

(２)中心在Ｃ(a,0)，半径为a；

(３)中心在(a,/2)，半径为a；

(４)中心在Ｃ(

0,

０)，半径为r。

＝r

＝2acos

＝2asin

2+

0

2-2

0cos(-

０)=r2Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.你可以用极坐标方程直接来求吗？Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.练习以极坐标系中的点(1,1)为圆心，1为半径的圆的方程是CEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.题组练习2Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.（）A、双曲线B、椭圆C、抛物线D、圆DEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.（）CEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.ONMC(4,0)Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.直线的极坐标方程Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.答：与直角坐标系里的情况一样，求曲线的极坐标方程就是找出曲线上动点Ｐ的坐标

与

之间的关系，然后列出方程(,)=0

，再化简并讨论。怎样求曲线的极坐标方程？Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.例题1：求过极点，倾角为的射线的极坐标方程。oMx﹚分析：如图，所求的射线上任一点的极角都是，其极径可以取任意的非负数。故所求直线的极坐标方程为新课讲授Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.1、求过极点，倾角为的射线的极坐标方程。易得思考：2、求过极点，倾角为的直线的极坐标方程。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.和前面的直角坐标系里直线方程的表示形式比较起来，极坐标系里的直线表示起来很不方便，要用两条射线组合而成。原因在哪？为了弥补这个不足，可以考虑允许极径可以取全体实数。则上面的直线的极坐标方程可以表示为或Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.例题2、求过点A(a,0)(a>0)，且垂直于极轴的直线L的极坐标方程。解：如图，设点为直线L上除点A外的任意一点，连接OMox﹚AM在中有即可以验证，点A的坐标也满足上式。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.求直线的极坐标方程步骤1、根据题意画出草图；2、设点是直线上任意一点；3、连接MO；4、根据几何条件建立关于的方程，并化简；5、检验并确认所得的方程即为所求。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.练习：设点P的极坐标为A，直线过点P且与极轴所成的角为,求直线的极坐标方程。解：如图，设点为直线上异于的点连接OM，﹚oMxP(A)在中有即显然A点也满足上方程。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.例题3设点P的极坐标为，直线过点P且与极轴所成的角为,求直线的极坐标方程。oxMP﹚﹚Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.解：如图，设点点P外的任意一点，连接OM为直线上除则由点P的极坐标知设直线L与极轴交于点A。则在由正弦定理得显然点P的坐标也是它的解。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.OHMAEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.A、两条相交的直线B、两条射线C、一条直线D、一条射线Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.()BEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.()CEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.()BEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.OXABEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011A