A-maths!

(a) Express the equation in the form (x-a)^2 + (y-b)^2 = r^2 where (a,b) is the centre.

C1:
x^2 + y^2 + 12x - 6y + 20 = 0
x^2+12x+36 + y^2-6y+9 = 25
(x+6)^2 + (y-3)^2 = 5^2
Therefore Centre O1=(-6, 3), Radius = 5

C2:
x^2 + y^2 - 12x + 12y - 28 = 0
x^2-12x+36 + y^2+12y+36 = 100
(x-6)^2 + (y+6)^2 = 10^2
Therefore Centre O2=(6, -6), Radius = 10

(b) (i)
C1 - C2:
24x - 18y + 48 = 0
4x - 3y + 8 = 0

(b) (ii)
From (b)(i),
y = (4x + 8) / 3
Put this into C1,

x^2 + (4x + 8)^2 / 9 + 12x - 6 (4x + 8) / 3 + 20 = 0
9x^2 + 16x^2 + 64x + 64 + 108x - 72x - 144 + 180 = 0
25x^2 + 100x + 100 = 0
x^2 + 4x + 4 = 0
x = -2

4(-2) - 3y + 8 = 0
y = 0

Therefore the intersection point is (-2, 0).

x^2+y^2+12x-6y+20=0
x^2+y^2+12x-6y+36+9-25=0
(x+6)^2+(y-3)^2=5^2

O=(-6,3) r=5

x^2+y^2-12x+12y-28=0
x^2+y^2-12x+12y+36+36-100=0
(x-6)^2+(y+6)^2=10^2
O=(6,-6) r=10