www.4828.com:高中数学《4.2.2圆与圆的位置关系》课件 新人教A版必修2_图文

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4.2.2 圆与圆的位置关系

问题提出

1.点与圆、直线与圆的位置关系 有哪几种?如何判定这些位置关系? 2.圆与圆的位置关系有哪几种? 如何根据圆的方程判断圆与圆的位 置关系,我们将进一步探究.

知识探究(一):圆与圆的位置关系

思考1:两个大小不等的圆,其位置关 系有内含、内切、相交、外切、外离 等五种,在平面几何中,这些位置关 系是如何判定的?
d d

d

d

d

思考2:已知两圆 C1:x2+y2+D1x+E1y+F1=0和 C2:x2+y2+D2x+E2y+F2=0,用上述方法 判断两个圆位置关系的操作步骤如 何? 1.将两圆的方程化为标准方程;
2.求两圆的圆心坐标和半径R、r; 3.求两圆的圆心距d; 4.比较d与R-r,R+r的大小关系:

若d<|R-r|,则两圆内含; 若d=|R-r|,则两圆内切; 若|R-r|<d<R+r,则两圆相交; 若d=R+r,则两圆外切; 若d>R+r,则两圆外离. 思考3:能否根据两个圆的公共点个 数判断两圆的位置关系?

思考4:两个大小相等的圆的位置关 系有哪几种?

知识探究(二):相交圆的交线方程

思考1:已知两圆 C 1: x2+y2+D1x+E1y+F1=0和 C 2: x2+y2+D2x+E2y+F2=0, 则方程
x2+y2+D1x+E1y+F1-(x2+y2+D2x+E2y+F2)=0

表示的图形是什么?

思考2:若两圆 C1: x2+y2+D1x+E1y+F1=0 和 C 2: x2+y2+D2x+E2y+F2=0相交, M (x0,y0)为一个交点, 则 点M(x0,y0)在直线
(D1-D2)x+(E1-E2)y+F1-F2=0上吗?

思考3:若两圆 C1 : 2+y2+D x+E y+F =0和 x C 2: 1 1 1 x2+y2+D2x+E2y+F2=0相交, 则其 公共弦所在直线的方程是 (D1-D2)x+(E1-E2)y+F1-F2=0,那么过 交点的圆系方程是什么?
m(x2+y2+D1x+E1y+F1)+n(x2+y2+D2x+E2y+F2)=0

思考4:若两圆 C1:x2+y2+D1x+E1y+F1=0和 C2:x2+y2+D2x+E2y+F2=0相切, 则方程 (D1-D2)x+(E1-E2)y+F1-F2=0表示的 直线是什么?若两圆相离呢?

理论迁移

例1 已知圆C1:x2+y2+2x+8y- 2+y2-4x-4y-2=0, 8=0,圆C2:x 判断圆C1与圆C2的位置关系. 若相 交,求两圆的公共弦所在的直线方 程.
x2+y2-6x-4=0

例2 已知一个圆的圆心为M(2,1), 且与圆C:x2+y2-3x=0相交于A、B两 点,若圆心M到直线AB的距离为 5 ,求 圆M的方程.
A DC B M

x2+y2-4x-2y-1=0

作业:
P132习题4.2A组:4,6,9,10,11.


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