Question

Prove that the centre of a circle which cuts orthogonally two given circles lies on the radical axes of the two circles.

Answer 1

be the two given circles. Suppose a circle S ≡ x² + y² + 2gx + 2fy + c = 0 cuts orthogonally S' = 0 and S'' = 0. Therefore, by Theorem 3.17, we have

2gg' +2ff' = c + c'

and  2gg'' + 2ff'' = c + c''

Therefore, 2(g' - g'')g + 2(f' - f '')f = c' − c''. That is, the radical axis S' − S'' = 0 of S' = 0 and S' = 0 passes through (−g, −f).