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1、读书之法,在循序而渐进,熟读而精思坐标系与参数方程知识点1平面直角坐标系中的坐标伸缩变换设点 P(x,y)是平面直角坐标系中的任意一点,在变换(0):(0)xxyy的作用下,点 P(x,y)对应到点(,)P x y,称为平面直角坐标系中的坐标伸缩变换,简称伸缩变换.2.极坐标系的概念(1)极坐标系如图所示,在平面内取一个定点O,叫做极点,自极点O引一条射线Ox,叫做极轴;再选定一个长度单位,一个角度单位(通常取弧度)及其正方向(通常取逆时针方向),这样就建立了一个极坐标系.注:极坐标系以角这一平面图形为几何背景,而平面直角坐标系以互相垂直的两条数轴为几何背景;平面直角坐标系内的点与坐标能建立一

2、一对应的关系,而极坐标系则不可.但极坐标系和平面直角坐标系都是平面坐标系.(2)极坐标设 M是平面内一点,极点O与点 M的距离|OM|叫做点 M的极径,记为;以极轴Ox为始边,射线OM为终边的角xOM叫做点 M的极角,记为.有序数对(,)叫做点 M的极坐标,记作(,)M.一般地,不作特殊说明时,我们认为0,可取任意实数.特别地,当点M在极点时,它的极坐标为(0,)(R).和直角坐标不同,平面内一个点的极坐标有无数种表示.如果规定0,02,那么除极点外,平面内的点可用唯一的极坐标(,)表示;同时,极坐标(,)表示的点也是唯一确定的.3.极坐标和直角坐标的互化(1)互化背景:把直角坐标系的原点作为

3、极点,x 轴的正半轴作为极轴,并在两种坐标系中取相同的长度单位,如图所示:(2)互化公式:设M是坐标平面内任意一点,它的直角坐标是(,)x y,极坐标是(,)(0),于是极坐标与直角坐标的互化公式如表:点M直角坐标(,)x y极坐标(,)互化公式cossinxy222tan(0)xyyxx在一般情况下,由tan确定角时,可根据点M所在的象限最小正角.读书之法,在循序而渐进,熟读而精思4.常见曲线的极坐标方程曲线图形极坐标方程圆心在极点,半径为r的圆(02)r圆心为(,0)r,半径为r的圆2 cos()22r圆心为(,)2r,半径为r的圆2 sin(0)r过极点,倾斜角为的直线(1)()()RR

4、或(2)(0)(0)和过点(,0)a,与极轴垂直的直线cos()22a过点(,)2a,与极轴平行的直线sin(0)a注:由于平面上点的极坐标的表示形式不唯一,即(,),(,2),(,),(,),都表示同一点的坐标,这与点的直角坐标的唯一性明显不同.所以对于曲线上的点的极坐标的多种表示形式,只要求至少有一个能满足极坐标方程即可.例如对于极坐标方程,点(,)44M可以表示为5(,2)(,2),444444或或(-)等多种形式,其中,只有(,)44的极坐标满足方程.二、参数方程1.参数方程的概念一般地,在平面直角坐标系中,如果曲线上任意一点的坐标,x y都是某个变数t的函数()()xf tyg t,

5、并且对于t的每一个允许值,由方文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10

6、T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P

7、4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7

8、O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W

9、10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O

10、2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9

11、S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3读书之法,在循序而渐进,熟读而精思程组所确定的点(,)M x y都在这条曲线上,那么方程就叫做这条曲线的参数方程,联系变数,x y的变数t叫做参变数,简称参数,相对于参数方程而言,直接给出点的坐标间关系的方程叫做普通方程.2.参数方程

12、和普通方程的互化(1)曲线的参数方程和普通方程是曲线方程的不同形式,一般地可以通过消去参数而从参数方程得到普通方程.(2)如果知道变数,x y中的一个与参数t的关系,例如()xf t,把它代入普通方程,求出另一个变数与参数的关系()yg t,那么()()xf tyg t就是曲线的参数方程,在参数方程与普通方程的互化中,必须使,x y的取值范围保持一致.注：普通方程化为参数方程，参数方程的形式不一定唯一。应用参数方程解轨迹问题，关键在于适当地设参数，如果选用的参数不同，那么所求得的曲线的参数方程的形式也不同。3圆的参数如图所示，设圆O的半径为r，点M从初始位置0M出发，按逆

13、时针方向在圆O上作匀速圆周运动，设(,)M x y，则cos()sinxryr为参数。这就是圆心在原点O，半径为r的圆的参数方程，其中的几何意义是0OM转过的角度。圆心为(,)a b，半径为r的圆的普通方程是222()()xaybr，它的参数方程为：cos()sinxarybr为参数。4椭圆的参数方程以坐标原点O为中心，焦点在x轴上的椭圆的标准方程为22221(0),xyabab其参数方程为cos()sinxayb为参数，其中参数称为离心角；焦点在y轴上的椭圆的标准方程是22221(0),yxabab其参数方程为cos(),sinxbya为参数其中参数仍为离心角，通常规定参数的范

14、围为0，2）。注：椭圆的参数方程中，参数的几何意义为椭圆上任一点的离心角，要把它和这一点的旋转角区分开来，除了在四个顶点处，离心角和旋转角数值可相等外（即在0到2的范围内），在其他任何一点，两个角的数值都不相等。但当02时，相应地也有02，在其他象限内类似。5双曲线的参数方程以坐标原点O为中心，焦点在x轴上的双曲线的标准议程为22221(0,0),xyabab其参数方程为sec()tanxayb为参数，其中30,2),.22且文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编

15、码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1

16、 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3

17、ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文

18、档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2

19、H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F

20、3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T

21、3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3读书之法,在循序而渐进,熟读而精思焦点在y轴上的双曲线的标准方程是22221(0,0),yxabab其参数方程为cot(0,2).cscxbeya为参数，其中且以上参数都是双曲线上任意一点的离心角。6抛物线的参数方程以坐标原点为顶点，开口向右的抛物线22(0)ypx p的参数方程为22().2xpttypt为参数7直线的参数方程经过点000(,)Mxy，倾斜角为()2的直线l的普通方程是00tan(),yyxx而过000(,)Mx

22、y，倾斜角为的直线l的参数方程为00cossinxxtyyt()t为参数。注：直线参数方程中参数的几何意义：过定点000(,)Mxy，倾斜角为的直线l的参数方程为00cossinxxtyyt()t为参数，其中t表示直线l上以定点0M为起点，任一点(,)M x y为终点的有向线段0M M的数量，当点M在0M上方时，t0；当点M在0M下方时，t0；当点M与0M重合时，t=0。我们也可以把参数t理解为以0M为原点，直线l向上的方向为正方向的数轴上的点M的坐标，其单位长度与原直角坐标系中的单位长度相同。文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O

23、2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9

24、S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P

25、1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH

26、2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6

27、B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T

28、1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3文档编码:CH2O2P4H2H1 HO6B9S7O4F3 ZX4T1P1W10T3

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