高中數(shù)學(xué)的對(duì)稱(chēng)問(wèn)題

對(duì)稱(chēng)問(wèn)題2021/5/91引入思考：1.已知點(diǎn)A(-2,3)，如何求點(diǎn)A關(guān)于點(diǎn)P（1，2） 的對(duì)稱(chēng)點(diǎn)B？2.已知直線(xiàn)a:x+y-1=0，如何求直線(xiàn)a關(guān)于P(1,2)的對(duì)稱(chēng)直線(xiàn)b？3.已知點(diǎn)A(-2,3)，如何求點(diǎn)A關(guān)于直線(xiàn)x+y-1=0的對(duì)稱(chēng)點(diǎn)E？4.已知直線(xiàn)a:x-y+1=0，如何求直線(xiàn)a關(guān)于直線(xiàn)x+2y-1=0的對(duì)稱(chēng)直線(xiàn)l？2021/5/92知識(shí)回顧：2021/5/93（1）點(diǎn)對(duì)稱(chēng)（中心對(duì)稱(chēng)）：①點(diǎn)與點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)：點(diǎn)A(x1,y1)關(guān)于P(m,n)對(duì)稱(chēng)的點(diǎn)A＇的坐標(biāo)為：(2m-x1,2n-y1).②線(xiàn)與線(xiàn)關(guān)于點(diǎn)對(duì)稱(chēng)：曲線(xiàn)f(x,y)=0關(guān)于點(diǎn)P(m,n)對(duì)稱(chēng)的曲線(xiàn)方程為：f(2m-x,2n-y)=0.A(x1,y1)P(m,n)(2m-x1,2n-y1)A＇f(x,y)=0M(x,y)P(m,n)M′

(2m-x,2n-y)f(2m-x,2n-y)=02021/5/941.點(diǎn)關(guān)于點(diǎn)的對(duì)稱(chēng)設(shè)C（x，y），則由中點(diǎn)公式得解析：典例解析關(guān)于點(diǎn)對(duì)稱(chēng)的問(wèn)題即中心對(duì)稱(chēng)問(wèn)題，關(guān)鍵在于抓住兩對(duì)稱(chēng)點(diǎn)被中心平分這一性質(zhì)；二．對(duì)稱(chēng)問(wèn)題的處理方法2021/5/952.直線(xiàn)關(guān)于點(diǎn)的對(duì)稱(chēng)例2、求直線(xiàn)y=3x–4關(guān)于點(diǎn)P(2,–1)的對(duì)稱(chēng)直線(xiàn)方程.P(2,–1)xyOy=3x–4分析一:將直線(xiàn)的對(duì)稱(chēng)轉(zhuǎn)化為直線(xiàn)上的點(diǎn)的對(duì)稱(chēng).還可以有什么方法？2021/5/96（2）軸對(duì)稱(chēng)①點(diǎn)與點(diǎn)關(guān)于直線(xiàn)對(duì)稱(chēng)：

求點(diǎn)M關(guān)于直線(xiàn)l對(duì)稱(chēng)的點(diǎn)M′的方法：M(x1,y1)M′(x,y)l由線(xiàn)段MM′的中點(diǎn)坐標(biāo)滿(mǎn)足l的方程(中點(diǎn)在l上)及kMM′·k

l=-1(MM′⊥l)建立方程組求得。設(shè)l:Ax+By+C=0，M(x1,y1)，M′(x,y)，則有：，由此解得x,y。即：2021/5/97C1lC2M(x,y)M＇(x1,y1)②線(xiàn)與線(xiàn)關(guān)于直線(xiàn)對(duì)稱(chēng)：求曲線(xiàn)C1:f(x,y)=0關(guān)于直線(xiàn)l對(duì)稱(chēng)的曲線(xiàn)C2的方法：設(shè)C1:f(x,y)=0，l:Ax+By+C=0,步驟：在C2上任取一點(diǎn)M(x,y),則點(diǎn)M關(guān)于的對(duì)稱(chēng)點(diǎn)M′(x1,y1)

必在C1上;

將上面求得的x1,y1代入C1方程，整理即得的C2方程。

由解出x1,y1(均是用x,y表示的)；2021/5/98（3）幾種特殊的對(duì)稱(chēng)：③曲線(xiàn)f(x,y)=0關(guān)于直線(xiàn)y=x+b對(duì)稱(chēng)的曲線(xiàn)方程為_(kāi)___________;

關(guān)于直線(xiàn)y=-x+b對(duì)稱(chēng)的曲線(xiàn)方程為_(kāi)_____________.①點(diǎn)P(x,y)關(guān)于下列點(diǎn)或線(xiàn)的對(duì)稱(chēng)點(diǎn)分別為：關(guān)于原點(diǎn):______________;關(guān)于x軸:______________;關(guān)于y軸:______________;關(guān)于直線(xiàn)y=x:__________;關(guān)于直線(xiàn)y=-x:__________;關(guān)于直線(xiàn)x=a:__________.②曲線(xiàn)f(x,y)=0關(guān)于下列點(diǎn)或直線(xiàn)對(duì)稱(chēng)的曲線(xiàn)方程分別為：關(guān)于原點(diǎn):________________;關(guān)于x軸:____________;關(guān)于y軸:________________;關(guān)于直線(xiàn)y=x:____________;關(guān)于直線(xiàn)y=-x:____________;關(guān)于直線(xiàn)x=a:____________.(-x,-y)(x,-y)(-x,y)(y,x)(-y,-x)(2a-x,y)f(-x,-y)=0f(x,-y)=0f(-x,y)=0f(y,x)=0f(-y,-x)=0f(2a-x,y)=0f(y-b,x+b)=0f(-y+b,-x+b)=02021/5/99對(duì)稱(chēng)問(wèn)題是中學(xué)數(shù)學(xué)常見(jiàn)的一類(lèi)數(shù)學(xué)問(wèn)題?？煞譃閮深?lèi)：Ⅰ抽象函數(shù)的對(duì)稱(chēng)問(wèn)題相關(guān)的結(jié)論有：一、對(duì)稱(chēng)問(wèn)題的類(lèi)型2021/5/910Ⅱ解析幾何中的對(duì)稱(chēng)問(wèn)題

這是近幾年高考中常見(jiàn)的題型之一。其思想是一種常用的數(shù)學(xué)思想方法，是近幾年高考考查的熱點(diǎn)問(wèn)題之一。相關(guān)結(jié)論有：2021/5/911例1、直線(xiàn)y=x-1關(guān)于點(diǎn)(2,3)對(duì)稱(chēng)的直線(xiàn)方程為_(kāi)__________;

關(guān)于直線(xiàn)y=2x對(duì)稱(chēng)的直線(xiàn)方程為_(kāi)________________.例2、直線(xiàn)2x+y-3=0關(guān)于直線(xiàn)x-y-1=0對(duì)稱(chēng)的直線(xiàn)是__________;

關(guān)于直線(xiàn)x+y-2=0對(duì)稱(chēng)的直線(xiàn)是________________.例3、已知點(diǎn)A(-3,5),B(2,15),直線(xiàn)l:3x-4y+4=0,在l上求一點(diǎn)P，