Radical axis:

The radical axis of 2 circles is the locus of a point from which the tangent segments to the two circles are of equal length.

Equation to the Radical Axis

Consider S ≡ x² + y² + 2gx + 2fy + c = 0

and S' ≡ x² + y² + 2g'x + 2f'y + c' =0, then S-S'=0 gives the equation of the Radical Axis to the 2w circles that is 2x(g - g') + 2y(f - f') + c - c' = 0.

Note:

1. If S = 0 and S' = 0 intersect in real and distinct points then S - S' = 0 is the equation of common chord of the 2 circles.

2. If S' = 0 and S = 0 touch each other, then S - S' = 0 is equation of the common tangent to the 2 circles at the point of contact.

3. To find out the equation of the radical axis of 2 circles, first make the coefficients of x² and y² in equation of the 2 circles equal to unity.

4. The radical axis of 2 circles is perpendicular to the line joining their centres.

5. The radical axis of 3 circles taken in pairs meet at a point, called as radical centre of the circles. Coordinates of the radical centre can be found by solving equations S1=S2=S3=0.

6. The radical centre of 3 circles described on the sides of the triangle as diameters is the orthocentre of the triangle.

7. If 2 circles cut a 3^rd circle orthogonally, then the radical axis of the 2 circles pass through the centre of the 3^rd circle.

or the locus of the centre of a circle cutting 2 given circles orthogonally is the radical axis of the 2 circles.

8. The radical axis of 2 circles will bisect their common tangents.

Illustration:   If circle x²+y²+2a₁x+2b₁y+c₁=0 bisects the circumference of x²+y²+2a₂x+2b₂y+c₂=0, then show that 2a₂ (a₁ - a₂)x + 2b₂(b₁ - b₂)y + c₁ - c₂ = 0

Solution: Clearly centre of 2^nd circle i.e., (-a₂, -b₂) should lie on common chord of a circles that is, on the line 2 (a₁ - a₂)x + 2(b₁ - b₂)y + c₁ - c₂ = 0

   Thus 2a₂ (a₁ - a₂)x + 2b₂(b₁ - b₂)y + c₁ - c₂ = 0

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