amaths circle

The circle: x2 + y2 = 1
centre at O(0, 0) and radius = 1

(x-10)2 + y2 = 16.
centre at Q(10, 0) and radius = √16 = 4

Let the centre of the required centre be C(a,b) with radius r,
and the circle passes through the point P(1, -14).

圖片參考：http://i277.photobucket.com/albums/kk75/uncle_michael/080708_mat01.jpg?t=1215504581

CO = r + 1
√[(a - 0)2 + (b - 0)2] = r + 1
r = -1 + √(a2 + b2) ...... (1)

CQ = r + 4
√[(a - 10)2 + (b - 0)2] = r + 4
r = -4 + √(a2 + b2 - 20a + 100) ...... (2)

CP = r
√[(a - 1)2 + (b + 14)2] = r
r = √(a2 + b2 - 2a + 28b + 197) ...... (3)

(1) = (2)
-1 + √(a2 + b2) = -4 + √(a2 + b2 - 20a + 100)
3 + √(a2 + b2) = √(a2 + b2 - 20a + 100)
9 + a2 + b2 + 6√(a2 + b2) = a2 + b2 - 20a + 100
6√(a2 + b2) = -20a + 91 ...... (4)

(1) = (3)
-1 + √(a2 + b2) = √(a2 + b2 - 2a + 28b + 197)
[-1 + √(a2 + b2)]2 = [√(a2 + b2 - 2a + 28b + 197)]2
1 + a2 + b2 - 2√(a2 + b2) = a2 + b2 - 2a + 28b + 197
-2√(a2 + b2) = -2a + 28b + 196
6√(a2 + b2) = 6a - 84b - 588 ...... (5)

(4) = (5)
- 20a + 91 = 6a - 84b - 588
84b = 26a - 679
b = (26a - 679)/84 ...... (6)

Subst. (6) into (4)
6√{a2 + [(26a - 679)/84]2} = -20a + 91
36{a2 + (26a - 679)2/7056} = (-20a + 91)2
36[7056a2 + (26a - 679)2] = 7056 (-20a + 91)2
36[7056a2 + (26a - 679)2] = 7056 (-20a + 91)2
[7056a2 + (26a - 679)2] = 196(-20a + 91)2
7056a2 + 676a2 - 35308a + 461041 = 196(400a2 - 3640a + 8281)
7056a2 + 676a2 - 35308a + 461041 = 78400a2 - 713440a + 1623076
70668a2 - 678132a + 1162035 = 0
a = [678132 √(6781322 - 4 x 70668 x 1162035)] / (2 x 70668)
Hence, a = 7.362 ororor a = 2.233

When a = 7.362, subst. into (6)
b = (26 x 7.362 - 679)/84
b = -5.805
(Rejected because the circle touches internally the circle with centre Q.)

When a = 2.233, subst. into (6)
b = (26 x 2.233 - 679)/84
b = -7.392

Ans: The centre of the required circle is (2.233, -7.392)
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