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1、新课标高中数学圆的方程典型例题类型一：圆的方程例 1 求过两点)4,1(A、)2,3(B且圆心在直线0y上的圆的标准方程并判断点)4,2(P与圆的关系分析：欲求圆的标准方程，需求出圆心坐标的圆的半径的大小，而要判断点P与圆的位置关系，只须看点P与圆心的距离和圆的半径的大小关系，若距离大于半径，则点在圆外；若距离等于半径，则点在圆上；若距离小于半径，则点在圆内解法一：（待定系数法）设圆的标准方程为222)()(rbyax圆心在0y上，故0b圆的方程为222)(ryax又该圆过)4,1(A、)2,3(B两点22224)3(16)1(rara解之得：1a，202r所以所求圆的方程为20)1(22yx
2、解法二：（直接求出圆心坐标和半径）因为圆过)4,1(A、)2,3(B两点，所以圆心C必在线段AB的垂直平分线l上，又因为13124ABk，故l的斜率为1，又AB的中点为)3,2(，故AB的垂直平分线l的方程为：23xy即01yx又知圆心在直线0y上，故圆心坐标为)0,1(C半径204)11(22ACr故所求圆的方程为20)1(22yx又点)4,2(P到圆心)0,1(C的距离为rPCd254)12(22点P在圆外说明：本题利用两种方法求解了圆的方程，都围绕着求圆的圆心和半径这两个关键的量，然后根据圆心与定点之间的距离和半径的大小关系来判定点与圆的位置关系，若将点换成直线又该如何来判定直线与圆的位
3、置关系呢？例 2 求半径为4，与圆042422yxyx相切，且和直线0y相切的圆的方程分析：根据问题的特征，宜用圆的标准方程求解解：则题意，设所求圆的方程为圆222)()(rbyaxC：圆C与直线0y相切，且半径为4，则圆心C的坐标为)4,(1aC或)4,(2aC又已知圆042422yxyx的圆心A的坐标为)1,2(，半径为3若两圆相切，则734CA或134CA(1)当)4,(1aC时，2227)14()2(a，或2221)14()2(a(无 解)，故 可 得1022a所求圆方程为2224)4()1022(yx，或2224)4()1022(yx(2)当)4,(2aC时，2227)14()2(a
4、，或2221)14()2(a(无 解)，故622a所求圆的方程为2224)4()622(yx，或2224)4()622(yx说明：对本题，易发生以下误解：由 题 意，所 求 圆 与 直 线0y相 切 且 半 径 为4，则 圆 心 坐 标 为)4,(aC，且 方 程 形 如2224)4()(yax又圆042422yxyx，即2223)1()2(yx，其圆心为)1,2(A，半径为 3 若两圆相切，则34CA 故2227)14()2(a，解之得1022a 所以欲求圆的方程为2224)4()1022(yx，或2224)4()1022(yx上述误解只考虑了圆心在直线0y上方的情形，而疏漏了圆心在直线0y
5、下方的情形另外，误解中没有考虑两圆内切的情况也是不全面的例 3 求经过点)5,0(A，且与直线02yx和02yx都相切的圆的方程分析：欲确定圆的方程 需确定圆心坐标与半径，由于所求圆过定点A，故只需确定圆心坐标又圆与两已知直线相切，故圆心必在它们的交角的平分线上解：圆和直线02yx与02yx相切，圆心C在这两条直线的交角平分线上，又圆心到两直线02yx和02yx的距离相等文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X
6、1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J
7、3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X
8、1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J
9、3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X
10、1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J
11、3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X
12、1E1A6S35252yxyx两直线交角的平分线方程是03yx或03yx又圆过点)5,0(A，圆心C只能在直线03yx上设圆心)3,(ttCC到直线02yx的距离等于AC，22)53(532tttt化简整理得0562tt解得：1t或5t圆心是)3,1(，半径为5或圆心是)15,5(，半径为55所求圆的方程为5)3()1(22yx或125)15()5(22yx说明：本题解决的关键是分析得到圆心在已知两直线的交角平分线上，从而确定圆心坐标得到圆的方程，这是过定点且与两已知直线相切的圆的方程的常规求法例 4、设圆满足：(1)截y轴所得弦长为2；(2)被x轴分成两段弧，其弧长的比为1:3，在满足条件(
13、1)(2)的所有圆中，求圆心到直线02yxl：的距离最小的圆的方程分析：要求圆的方程，只须利用条件求出圆心坐标和半径，便可求得圆的标准方程满足两个条件的圆有无数个，其圆心的集合可看作动点的轨迹，若能求出这轨迹的方程，便可利用点到直线的距离公式，通过求最小值的方法找到符合题意的圆的圆心坐标，进而确定圆的半径，求出圆的方程解法一：设圆心为),(baP，半径为r则P到x轴、y轴的距离分别为b和a由题设知：圆截x轴所得劣弧所对的圆心角为90，故圆截x轴所得弦长为r2222br又圆截y轴所得弦长为2122ar又),(baP到直线02yx的距离为文档编码:CO3K6C4J3V5 HC8O9I2M6H4 Z
14、C8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6
15、C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 Z
16、C8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6
17、C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 Z
18、C8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6
19、C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 Z
20、C8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S352bad2225badabba4422)(242222baba1222ab当且仅当ba时取“=”号，此时55mind这时有1222abba11ba或11ba又2222br故所求圆的方程为2)1()1(22yx或2)1()1(22yx解法二：同解法一，得52baddba522225544dbdba将1222ba代入上式得：01554222dbdb上述方程有实根，故0)15(82d，55d将55d代入方程得1b又122
21、2ab1a文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V
22、5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E
23、1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V
24、5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E
25、1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V
26、5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E
27、1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3由12ba知a、b同号故所求圆的方程为2)1()1(22yx或2)1()1(22yx说明：本题是求点到直线距离最小时的圆的方程，若变换为求面积最小呢？类型二：切线方程、切点弦方程、公共弦方程例 5已知圆422yxO：，求过点42，P与圆O相切的切线解：点42，P不在圆O上，切线PT的直线方程可设为42xky根据rd21422kk解得43k所以424
28、3xy即01043yx因为过圆外一点作圆得切线应该有两条，可见另一条直线的斜率不存在易求另一条切线为2x说明：上述解题过程容易漏解斜率不存在的情况，要注意补回漏掉的解本题还有其他解法，例如把所设的切线方程代入圆方程，用判别式等于0 解决（也要注意漏解）还可以运用200ryyxx，求出切点坐标0 x、0y的值来解决，此时没有漏解例 6 两圆0111221FyExDyxC：与0222222FyExDyxC：相交于A、B两点，求它们的公共弦AB所在直线的方程分析：首先求A、B两点的坐标，再用两点式求直线AB的方程，但是求两圆交点坐标的过程太繁为了避免求交点，可以采用“设而不求”的技巧解：设两圆1C、
29、2C的任一交点坐标为),(00yx，则有：0101012020FyExDyx0202022020FyExDyx得：0)()(21021021FFyEExDDA、B的坐标满足方程0)()(212121FFyEExDD方程0)()(212121FFyEExDD是过A、B两点的直线方程文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3
30、文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8
31、O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3
32、文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8
33、O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3
34、文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8
35、O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3又过A、B两点的直线是唯一的两圆1C、2C的公共弦AB所在直线的方程为0)()(
36、212121FFyEExDD说明：上述解法中，巧妙地避开了求A、B两点的坐标，虽然设出了它们的坐标，但并没有去求它，而是利用曲线与方程的概念达到了目标从解题的角度上说，这是一种“设而不求”的技巧，从知识内容的角度上说，还体现了对曲线与方程的关系的深刻理解以及对直线方程是一次方程的本质认识它的应用很广泛例 7、过圆122yx外一点)3,2(M，作这个圆的两条切线MA、MB，切点分别是A、B，求直线AB的方程。练习：1求过点(3,1)M，且与圆22(1)4xy相切的直线l的方程 解：设切线方程为1(3)yk x，即310kxyk，圆心(1,0)到切线l的距离等于半径2，22|31|21kkk，解得
37、34k，切线方程为31(3)4yx，即34130 xy，当过点M的直线的斜率不存在时，其方程为3x，圆心(1,0)到此直线的距离等于半径2，故直线3x也适合题意。所以，所求的直线l的方程是34130 xy或3x2、过坐标原点且与圆0252422yxyx相切的直线的方程为解：设直线方程为kxy，即0ykx.圆方程可化为25)1()2(22yx，圆心为（2，-1），半径为210.依题意有2101122kk，解得3k或31k，直线方程为xy3或xy31.3、已知直线0125ayx与圆0222yxx相切，则a的值为.解：圆1)1(22yx的圆心为（1，0），半径为1，1125522a，解得8a或18a
38、.类型三：弦长、弧问题例 8、求直线063:yxl被圆042:22yxyxC截得的弦AB的长.文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J
39、3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X
40、1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J
41、3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X
42、1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J
43、3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X
44、1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3例 9、直线0323yx截圆422yx得的劣弧所对的圆心角为解：依题意得，弦心距3d，故弦长2222drAB，从而 OAB 是等边三角形，故截得的劣弧所对的圆心角为3AOB.例 10、求两圆0222yxyx和522yx的公共弦长类型四：直线与圆的位置关系例 11、已
45、知直线0323yx和圆422yx，判断此直线与已知圆的位置关系.例 12、若直线mxy与曲线24xy有且只有一个公共点，求实数m的取值范围.解：曲线24xy表示半圆)0(422yyx，利用数形结合法，可得实数m的取值范围是22m或22m.例 13 圆9)3()3(22yx上到直线01143yx的距离为1 的点有几个？分析：借助图形直观求解或先求出直线1l、2l的方程，从代数计算中寻找解答解法一：圆9)3()3(22yx的圆心为)3,3(1O，半径3r设圆心1O到直线01143yx的距离为d，则324311343322d如图，在圆心1O同侧，与直线01143yx平行且距离为1 的直线1l与圆有两
46、个交点，这两个交点符合题意又123dr与直线01143yx平行的圆的切线的两个切点中有一个切点也符合题意文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3
47、K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4
48、 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3
49、K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4
50、 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3K6C4J3V5 HC8O9I2M6H4 ZC8X1E1A6S3文档编码:CO3