1.3簡單曲線的極坐標方程Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.教學目標1、認識幾種圓的極坐標方程，比較它與直角坐標方程的異同。2、掌握求圓的極坐標方程的方法。3、能應用極坐標方程解決圓與直線的關系。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.教學重點：

求圓的極坐標方程的方法與步驟教學難點：極坐標方程是涉及長度與角度的問題，列方程實質是解直角或斜三角形問題，要使用舊的三角知識。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.答：與直角坐標系里的情況一樣，求曲線的極坐標方程就是建系－設點（點與坐標的對應）－列式（方程與坐標的對應）－化簡得方程f(,)=0

－說明求曲線的極坐標方程的步驟：Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.新課引入：熱身訓練：在平面直角坐標系中1、圓心坐標為(3,0)且半徑為3的圓方程為

（x-3）2+y2=92、圓心坐標為（0,3）且半徑為3的圓線方程為_______3、圓心在原點半徑為3的圓方程為_______

X2+(y-3)2=9X2+y2=9變式：將以上三個方程化為極坐標方程并畫出對應的圖形。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.直角坐標系極坐標系極坐標圖形1、圓心（3，0）半徑為3圓心（3，0）半徑為32、圓心（0，3）半徑為3圓心（3,／2)3、圓心（0，0）半徑為3圓心在極點，半徑為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.曲線的極坐標方程定義：如果曲線Ｃ上的點與方程f(,)=0有如下關系(１)曲線Ｃ上任一點的坐標(所有坐標中至少有一個)符合方程f(,)=0；(２)方程f(,)=0的所有解為坐標的點都在曲線Ｃ上。

則曲線Ｃ的方程是f(,)=0。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.題組練習1求下列圓的極坐標方程(１)中心在極點，半徑為r；

(２)中心在Ｃ(a,0)，半徑為a；

(３)中心在(a,/2)，半徑為a；

(４)中心在Ｃ(

0,

０)，半徑為r。

＝r

＝2acos

＝2asin

2+

0

2-2

0cos(-

０)=r2Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.你可以用極坐標方程直接來求嗎？Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.練習以極坐標系中的點(1,1)為圓心，1為半徑的圓的方程是CEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.題組練習2Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.（）A、雙曲線B、橢圓C、拋物線D、圓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.直線的極坐標方程Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.答：與直角坐標系里的情況一樣，求曲線的極坐標方程就是找出曲線上動點Ｐ的坐標

與

之間的關系，然后列出方程(,)=0

，再化簡并討論。怎樣求曲線的極坐標方程？Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.例題1：求過極點，傾角為的射線的極坐標方程。oMx﹚分析：如圖，所求的射線上任一點的極角都是，其極徑可以取任意的非負數。故所求直線的極坐標方程為新課講授Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.1、求過極點，傾角為的射線的極坐標方程。易得思考：2、求過極點，傾角為的直線的極坐標方程。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.和前面的直角坐標系里直線方程的表示形式比較起來，極坐標系里的直線表示起來很不方便，要用兩條射線組合而成。原因在哪？為了彌補這個不足，可以考慮允許極徑可以取全體實數。則上面的直線的極坐標方程可以表示為或Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.例題2、求過點A(a,0)(a>0)，且垂直于極軸的直線L的極坐標方程。解：如圖，設點為直線L上除點A外的任意一點，連接OMox﹚AM在中有即可以驗證，點A的坐標也滿足上式。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.求直線的極坐標方程步驟1、根據題意畫出草圖；2、設點是直線上任意一點；3、連接MO；4、根據幾何條件建立關于的方程，并化簡；5、檢驗并確認所得的方程即為所求。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.練習：設點P的極坐標為A，直線過點P且與極軸所成的角為,求直線的極坐標方程。解：如圖，設點為直線上異于的點連接OM，﹚oMxP(A)在中有即顯然A點也滿足上方程。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.例題3設點P的極坐標為，直線過點P且與極軸所成的角為,求直線的極坐標方程。oxMP﹚﹚Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.解：如圖，設點點P外的任意一點，連接OM為直線上除則由點P的極坐標知設直線L與極軸交于點A。則在由正弦定理得顯然點P的坐標也是它的解。Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.OHMAEvaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.Evaluationonly.CreatedwithAspose.Slidesfor.NET3.5ClientProfile.Copyright2004-2011AsposePtyLtd.A、兩條相交的直線B、兩條射線C、一條直線D、一條射線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