A.math family of circle

The common chord is x + y - 4 = 0 or y = 4 - x ...(1)
Sub into circle x^2 + y^2 - 4x - 8y + 16 = 0
x^2 + (4 - x)^2 - 4x - 8(4 - x) + 16 = 0
x^2 + 16 - 8x + x^2 - 4x - 32 + 8x + 16 = 0
2x^2 - 4x = 0
2x(x - 2) = 0
x = 0 or x = 2
x= 0 => y = 4
x = 2 => y = 2
The intersection points A & B are (0,4) and (2,2).
The smallest member circle is one with AB as diameter.
The centre of this circle = [(0+2)/2,(4+2)/2] = (1, 3)
The radius of the circle is √(1 + 1) = √2
The equation of the circle is (x - 1)^2 + (y - 3)^2 = 2
x^2 - 2x + 1 + y^2 - 6y + 9 = 2
x^2 + y^2 - 2x - 6y + 8 = 0

Equation of the family of circles is
x^2 + y^2 - 4x - 8y + 16 + k(x^2 + y^2 + 2x - 2y - 8) = 0
Rearranging we get
(1 + k)x^2 + (1 + k)y^2 + (2k - 4)x - (8 + 2k)y + (16 - 8k) = 0
so co-ordinates of the centre is
x = (4 - 2k)/2(1 + k) and y = (8 + 2k)/2(1 + k) or
x = (2 - k)/(1 + k) and y = (4 + k)/(1 + k).
The smallest circle in the family is the one with the common chord as diameter, that means the centre of the smallest circle is on the common chord, therefore,
(2-k)/(1 + k) + (4 + k)/(1 + k) = 4
2 - k + 4 + k = 4(1 + k)
6 = 4 + 4k
Sub. into the family, we get
3x^2/2 + 3y^2/2 - 3x - 9y + 12 = 0
3x^2 + 3y^2 - 6x - 18y + 24 = 0
x^2 + y^2 - 2x - 6y + 8 = 0.



2009-09-19 17:45:55 補充：
Missing: 6 = 4 + 4k, so k = 1/2.