解:不妨设C1,C2和C的半径分别为r1,r2,r(r1>r2) (1)当C1和C2相离时,即|C1C2|>r1+r2, (i)若C与C1,C2都外切, 则|CC1|=r1+r,|CC2|=r2+r, ∴|CC1|-|CC2|=r1-r2 若C与C1,C2都内切, 则|CC1|=r-r1,|CC2|=r-r2, ∴|CC2|-|CC1|=r1-r2; ∴||CC2|-|CC1||=r1-r2<|C1C2|,由双曲线的定义,C的圆心的轨迹是以C1,C2为焦点、实轴长为r1-r2的双曲线; (ii)若C与Cl内切,C2外切, 则|CC1|=r-r1,|CC2|=r2+r, ∴|CC2|-|CC1|=r1+r2; 若C与C1外切,C2内切, 则|CC1|=r+r1,|CC2|=r-r2, ∴|CC1|-|CC2|=r1+r2 ∴||CC2|-|CC1||=r1+r2<|C1C2|,由双曲线的定义,C的圆心的轨迹是以C1,C2为焦点、实轴长为r1+r2的双曲线; (2)当C1和C2外切时,即|C1C2|=r1+r2, (i)若C与C1,C2都外切, 则|CC1|=r1+r,|CC2|=r2+r, ∴|CC1|-|CC2|=r1-r2; 若C与C1,C2都内切, 则|CC1|=r-r1,|CC2|=r-r2, ∴|CC2|-|CC1|=r1-r2; ∴||CC2|-|CC1||=r1-r2<|C1C2|,由双曲线的定义,C的圆心的轨迹是以C1,C2为焦点、实轴长为r-r2的双曲线; |
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(ii)若C与C1内切,C2外切, 则|CC1|=r-r1,|CC2|=r2+r(或|CC1|=r1-r,|CC2|=r2+r)(如图1,2), ∴|CC2|-|CC1|=r1+r2(或|CC2|+|CC1|=r1+r2) 若C与C1外切,C2内切, 则|CC1|=r+r1,|CC2|=r-r2(或|CC1| =r+r1,|CC2|=r2-r), ∴|CC1|-|CC2|=r1+r2=|C1C2|(或|CC2|+|CC1|=r1+r2= |C1C2|), ∴C的圆心的轨迹是过C1,C2的直线(除直线与圆C1、C2的交点外); |
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(3)当C1和C2相交时, 即r1-r2<|C1C2| (i)若C与C1,C2都外切, 则|CC1|=r1+r,|CC2|=r2+r, ∴|CC1|-|CC2|=r1-r2 若C与C1,C2都内切, 则|CC1|=r-r1,|CC2|=r-r2(或|CC1| =r1-r,|CC2|=r2-r), ∴||CC2|-|CC1|| =r1-r2 ∴||CC2|-|CC1|| =r1-r2<|C1C2|, 由双曲线的定义,C的圆心轨迹是以C1,C2为焦点、实轴长为r1-r2的双曲线(圆C1,C2的交点除外); |
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(ii)若C与C1内切,C2外切(如图3), 则|CC1|=r1-r,|CC2|=r2+r, ∴|CC2|+|CC1|=r1+r2 若C与C1外切,C2内切, 则|CC1|=r+r1,|CC2|=r2-r, ∴|CC2|+|CC1|=r1+r2 ∴|CC2|+|CC1|=r1+r2>|C1C2|,由椭圆的定义,C的圆心的轨迹方程是以C1,C2为焦点、实轴长为r1+r2的椭圆(圆 C1、C2的交点除外); |
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(4)当C1和C2内切时, 即|C1C2| =r1-r2, (i)若C与C1,C2都外切, 则|CC1|=r1+r,|CC2|=r2+r, ∴|CC1|-|CC2|=r1-r2; 若C与C1,C2都内切, 则|CC1|=r-r1,|CC2|=r-r2(或|CC1| =r1-r,|CC2|=r-r2或|CC1|=r1-r,|CC2|=r2-r)(如图4, 5,6), ∴|CC2|-|CC1|=r1-r2(或|CC2|+|CC1|=r1-r2或|CC2|- |CC1|=r2-r1) ∴||CC2|-|CC1||=r1-r2=|C1C2|或|CC2|+|CC1|=r1-r2, ∴C的圆心的轨迹是过C1,C2的直线(除直线与圆C1 、C2的交点外); |
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(ii)若C与C1内切,C2外切, 则|CC1|=r1-r,|CC2|=r2+r, ∴|CC2|+|CC1|=r1+r2>|C1C2|, ∴C的圆心的轨迹是以C1,C2为焦点、实轴长为r1+r2的椭圆(两圆C1、C2的交点除外); |
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(5)当C1和C2内含时, 即|C1C2|<r1-r2, (i)若C与C1,C2都内切(如图7), 则|CC1|=r1-r,|CC2|=r-r2, ∴|CC2|+|CC1|=r1-r2>|C1C2|, ∴C的圆心的轨迹是以C1,C2为焦点、实轴长为r1-r2的椭圆; |
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(ii)若C与C1内切,C2外切, 则|CC1|=r1- r,|CC2|=r2+r, ∴|CC2|+|CC1|=r1+r2>|C1C2|, ∴C的圆心的轨迹是以C1,C2为焦点、实轴长为r1+r2的椭圆。 |
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