F.5 Maths equation of circle

Let O1 and O2 be the centres of the C1 and C2 respectively.

C1 : x² + y² - 16x - 12y + 75 = 0
Coordinates of O1 = (16/2, 12/2) = (8, 6)
Radius of C1 = √[(-16/2)² + (-12/2)² - 75] = 5

C2 : x² + y² + 2x + 12y - 63= 0
Coordinates of O 2 = (-2/2, -12/2) = (-1, -6)
Radius of C2 = √[(2/2)² + (12/2)² - (-63)]= 10

O1O2 = √[(8 + 1)² +(6 + 6)²] = 15
Radius of C1 + Radius of C2 = 15

The distance between the two centres isequal to the sum of the two radii.
Hence, the two circles touchexternally.

2015-03-18 20:06:13 補充：
Alternative method :

C1 : x² + y² - 16x - 12y + 75 = 0 ...... [1]
C2 : x² + y² + 2x + 12y - 63 = 0 ...... [2]

[2] - [1] :
18x + 24y - 138 = 0
3x + 4y - 23 = 0
x = (23 - 4y)/3 ...... [3]

2015-03-18 20:06:40 補充：
Put [3] into [2] :
[(23 - 4y)/3]² + y² + 2[(23 - 4y)/3] + 12y - 63 = 0
(23 - 4y)² + 9y² + 6(23 - 4y) + 108y - 567 = 0
529 - 184y + 16y² + 9y² + 138 - 24y + 108y - 567 = 0
25y² - 100y + 100 = 0
y² - 4y + 4 = 0
(y - 2)² = 0
y = 2 (double root)

2015-03-18 20:06:59 補充：
The two circles meet at only one point.
Hence, the two circles touch each other.