Let the line ax + by + c = 0 cut X, Y axes at A and B respectively co-ordinates of O are (0, 0) A are
(- c/a, 0) B are (0, - c/b)
Suppose the equation of the required circle is
x2 + y2 + 2gx + 2fy + c = 0
This circle passes through O(0, 0)
∴ c = 0
This circle passes through A(- c/a,0)
c2/a2 + 0 - 2gc/a = 0
2g.c/a = c2/a2
⇒ 2g = c/a ⇒ g = c/2a
The circle passes through
B(0, - c/b)
0 + c2/b2 + 0 - 2g c/b = 0
2f c/b = c2/b2
= 2g = c/b = f = c/2b
Equation of the circle through O, A, B is
x2 + y2 + cx/a + cy/b = 0
ab(x2 + y2) + (bx + ay) = 0
This is the equation of the circum circle of ∆OAB
