Question (Circles)

2009-02-15 7:51 am

Figure website: http://img223.imageshack.us/img223/600/0004lg1.jpg
C1 is a circle centred at the origin O and C2 is the circle x2 + y2 - 6x + 5 = 0. C1 and C2 touch each other externally as shown in the figure. The radii of C1 and C2 are 1 and 2 respectively.
(a) If A(s,t) is the centre which touches C1 and C2 internally, prove that t2 = 8s2 - 24s + 16.
(b) C3 is another circle with its centre at (0,8) and of the same radius as C1.
(i) If C1 and C3 both touch a circle internally whose centre is A(s,t), find the value of t.
(ii) Hence, find the equation of the circle in which C1, C2 and C3 touch ir internally.

回答 (1)

myisland8132

2009-02-15 8:28 am

✔ 最佳答案

是1995年hkcee題目的變種﹐不易。
(a)
The equation of C1: x^2+y^2=1
The equation of C2: x2 + y2 - 6x + 5 = 0
if(s,t) touches C1 and C2 externally and the required circle has radius r
√(s^2+t^2)=r+1
√[(s-3)^2+t^2)]=r+2
So √(s^2+t^2)+1=√[(s-3)^2+t^2)]
(s^2+t^2)+2√(s^2+t^2)+1=[(s-3)^2+t^2)]
√(s^2+t^2)=-3s+4
(s^2+t^2)=9s^2+24s+16
t2 = 8s2 - 24s + 16.
(b)(i)
The distance between O1O3 is 8
So t=(8-2)/2+1=3+1=4
(ii)
8s2 - 24s + 16=16
s^2-3s=0
s= 0 or 3
So centre A(0,4)
The equation of the circle in which C1, C2 and C3 touch externally is
(x-0)^2+(y-4)^2=9
x^2+y^2-8y+7=0

2009-02-15 00:30:58 補充：
應是reject s=3﹐因為若果s=3﹐個圓冇可能外touch 到c1

2009-02-15 00:34:12 補充：
外切两圆的连心线必经过它们的切点，且两个圆心之间的距离d（圆心距）等于两个圆的半径之和，即d=R+r 两圆外切

是此題的關鍵

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