Sunday, December 7, 2008

Radical axis of two circles

For two circles

Circle 1 (termed as S1) x² + y²+2g1x+2f1y+c = 0

Circle 2 (termed as S2) x² + y²+2g2x+2f2y+c = 0

Radical axis is
S1-S2 = 0

2x(g1-g2)+2y(f1-f2)+c1-c2 = 0

The equation has the same form at that of common chord of intersecting circles.

Properties of radical axis:
(i) The radical axis of two circles is always perpendicular to the line joining the centres.
(ii) The radical axes of three circles whose centres are non-collinear, taken in pairs, meet in a point. (This point is called radical centre)
(iii) The circle with centre at the radical centre and radius equal to the length of the tangent from it to any of the circles intersects all three circles orthogonally.

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