直線方程中的對(duì)稱(chēng)問(wèn)題

1、 對(duì)稱(chēng)問(wèn)題對(duì)稱(chēng)問(wèn)題中心對(duì)稱(chēng)問(wèn)題中心對(duì)稱(chēng)問(wèn)題點(diǎn)關(guān)于于點(diǎn)的對(duì)稱(chēng)點(diǎn)關(guān)于于點(diǎn)的對(duì)稱(chēng)線關(guān)于點(diǎn)的對(duì)稱(chēng)線關(guān)于點(diǎn)的對(duì)稱(chēng)軸對(duì)稱(chēng)問(wèn)題軸對(duì)稱(chēng)問(wèn)題點(diǎn)關(guān)于線的對(duì)稱(chēng)點(diǎn)關(guān)于線的對(duì)稱(chēng)線關(guān)于線的對(duì)稱(chēng)線關(guān)于線的對(duì)稱(chēng)（一）點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)（一）點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)一一.中心對(duì)稱(chēng)中心對(duì)稱(chēng)(關(guān)于點(diǎn)的對(duì)稱(chēng)關(guān)于點(diǎn)的對(duì)稱(chēng))1.點(diǎn)A（2，3）關(guān)于坐標(biāo)原點(diǎn)的對(duì)稱(chēng)點(diǎn)的坐標(biāo)_。(-2,-3)2.求點(diǎn)A（2，3）關(guān)于點(diǎn)B（-1，1）的對(duì)稱(chēng)點(diǎn)的坐_。 (-4,-1)3.求點(diǎn)A（2，3）關(guān)于任意一點(diǎn)B(a,b)的對(duì)稱(chēng)點(diǎn)的坐標(biāo)_。 （2a-2,2b-3)點(diǎn)關(guān)于點(diǎn)的對(duì)稱(chēng)點(diǎn)關(guān)于點(diǎn)的對(duì)稱(chēng)),(),(AyxAnmO)2 ,2(ynxm注：注： )0, 0(),(yx),(yx

2、 解題要點(diǎn)：解題要點(diǎn)：中點(diǎn)公式的運(yùn)用中點(diǎn)公式的運(yùn)用P(2,1)xyOy=3x4 求直線求直線y=3x4關(guān)于點(diǎn)關(guān)于點(diǎn)P(2,1)的對(duì)稱(chēng)直線方程的對(duì)稱(chēng)直線方程.（二）直線關(guān)于點(diǎn)的對(duì)稱(chēng)（二）直線關(guān)于點(diǎn)的對(duì)稱(chēng)P(2,1)xyOy=3x4求直線求直線y=3x4關(guān)于點(diǎn)關(guān)于點(diǎn)P(2,1)的對(duì)稱(chēng)直線方程的對(duì)稱(chēng)直線方程. 求直線求直線y=3x4關(guān)于點(diǎn)關(guān)于點(diǎn)P(2,1)的對(duì)稱(chēng)直線方程的對(duì)稱(chēng)直線方程.( , )M x yP設(shè)對(duì)稱(chēng)直線上任一點(diǎn)，則其關(guān)于 的對(duì)稱(chēng)點(diǎn)P(2,1)xyOy=3x4法三：分析一法三：分析一: : 將直線的對(duì)稱(chēng)轉(zhuǎn)化為直線上的點(diǎn)的對(duì)稱(chēng)將直線的對(duì)稱(chēng)轉(zhuǎn)化為直線上的點(diǎn)的對(duì)稱(chēng). .3100 xy化簡(jiǎn)得31

3、00.xy所求直線方程是上在直線43)2,4(Nxyyx4)4(32xy直線關(guān)于點(diǎn)對(duì)稱(chēng)直線關(guān)于點(diǎn)對(duì)稱(chēng)法二：法二：利用點(diǎn)到直線的距離利用點(diǎn)到直線的距離 l 1 / l 2且且P到兩直線到兩直線 等距。等距。主要方法：主要方法：法一：法一：轉(zhuǎn)化成求點(diǎn)關(guān)于點(diǎn)的對(duì)稱(chēng)轉(zhuǎn)化成求點(diǎn)關(guān)于點(diǎn)的對(duì)稱(chēng) 求點(diǎn)A（3，2）關(guān)于下列直線對(duì)稱(chēng)點(diǎn)的坐標(biāo)：二二.軸對(duì)稱(chēng)軸對(duì)稱(chēng)(即關(guān)于直線的對(duì)稱(chēng)即關(guān)于直線的對(duì)稱(chēng))(一一)點(diǎn)關(guān)于直線的對(duì)稱(chēng)點(diǎn)關(guān)于直線的對(duì)稱(chēng):0.（2,3）.（-2,-3）.（3,-2）.（3,2）.（-3，2）yxyx （法一）：（法一）：直線直線ABl, 直線直線AB過(guò)點(diǎn)（過(guò)點(diǎn)（-7，1）直線直線AB的方程為的方程為

4、y-1=- y-1=- （x+7x+7） 即即x+2y+5=0 x+2y+5=021052052yxyx由 解得 13xy即即AB的中點(diǎn)為（的中點(diǎn)為（1，-3） ，又，又A（-7，1）由中點(diǎn)坐標(biāo)公式得由中點(diǎn)坐標(biāo)公式得B的坐標(biāo)為（的坐標(biāo)為（9，-7）.例例. 求點(diǎn)求點(diǎn)A(-7,1)關(guān)于直線關(guān)于直線l:2x-y-5=0的對(duì)稱(chēng)點(diǎn)的對(duì)稱(chēng)點(diǎn)B的坐標(biāo)的坐標(biāo).例例.求點(diǎn)求點(diǎn)A(-7,1)關(guān)于直線關(guān)于直線l:2x-y-5=0的對(duì)稱(chēng)點(diǎn)的對(duì)稱(chēng)點(diǎn)B的坐標(biāo)的坐標(biāo).（法二）：設(shè)（法二）：設(shè)B(m,n）由點(diǎn)關(guān)于直線對(duì)稱(chēng)的定義知）由點(diǎn)關(guān)于直線對(duì)稱(chēng)的定義知: 線段線段ABl 即; =-1 2)7(1mn線段線段AB被直線被直

5、線l平分平分,即線段即線段AB的中點(diǎn)的中點(diǎn)21,27 nm在直線在直線l上上,故有故有 2 - -5=0 27m21n聯(lián)立聯(lián)立 解得解得m=9 n= -7B(9,-7)(法三法三) 設(shè)設(shè)B(m,n）由點(diǎn)關(guān)于直線對(duì)稱(chēng)的定義知）由點(diǎn)關(guān)于直線對(duì)稱(chēng)的定義知: 線段線段ABl 即; =-1 2)7(1mn 由題知：由題知：A，B兩點(diǎn)關(guān)于直線兩點(diǎn)關(guān)于直線l對(duì)稱(chēng)，則對(duì)稱(chēng)，則A,B兩點(diǎn)到兩點(diǎn)到直線直線l的距離是相等的，則：的距離是相等的，則：聯(lián)立聯(lián)立 解得解得m=9 n= -7B(9,-7)例例.求點(diǎn)求點(diǎn)A(-7,1)關(guān)于直線關(guān)于直線l:2x-y-5=0的對(duì)稱(chēng)點(diǎn)的對(duì)稱(chēng)點(diǎn)B的坐標(biāo)的坐標(biāo)點(diǎn)關(guān)于直線的對(duì)稱(chēng)點(diǎn)關(guān)于直線的對(duì)稱(chēng)),(0:AbaAcByAxl例例. 試求直線試求直線l1:x-y-2=0關(guān)于直線關(guān)于直線 l2:3x-y+3=0 對(duì)稱(chēng)的直線對(duì)稱(chēng)的直線l 的方程。的方程。解題要點(diǎn)：由線關(guān)于線對(duì)稱(chēng)轉(zhuǎn)化為點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)解題要點(diǎn)：由線關(guān)于線對(duì)稱(chēng)轉(zhuǎn)化為點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)思考：思考：若若l1/l2, 如何求如何求l1 關(guān)于關(guān)于l2的對(duì)稱(chēng)直線方程？的對(duì)稱(chēng)直線方程？C1lC2l1l2l1（二）直線關(guān)于直線的對(duì)稱(chēng)（二）直線關(guān)于直線的對(duì)稱(chēng)點(diǎn)點(diǎn)P(x,y)關(guān)于下列點(diǎn)或線的對(duì)稱(chēng)點(diǎn)分別為：關(guān)于下列點(diǎn)或線的對(duì)稱(chēng)點(diǎn)分別為：關(guān)于原點(diǎn)關(guān)于原點(diǎn):_; 關(guān)于關(guān)于x軸軸:_;關(guān)于關(guān)