Radical Axis of Two Circles: Theory & Construction

The radical axis of two circles is the line of all points with equal power for both circles. From any point on it, the tangent segments to the two circles have equal length.

This concept matters because it converts a geometric locus problem into simple algebra: subtracting the equations of two circles immediately gives the equation of their radical axis. It underlies the radical centre of three circles and many locus problems.

Key Concepts

• Power of a point: for point (x1, y1) and circle x² + y² + 2gx + 2fy + c = 0, power = x1² + y1² + 2gx1 + 2fy1 + c (equivalently OP² − r²).
• Radical axis: the locus of points with equal power for two circles; its equation is S1 − S2 = 0 once the x², y² coefficients are equalised.
• Normalisation: if the x² + y² coefficients differ, divide first so both equal 1 before subtracting.
• Position cases: for intersecting circles it is the common chord; for touching circles the common tangent; for separate circles a real line between them.
• Radical centre: the radical axes of three circles, taken in pairs, meet at a single point (the radical centre), which has equal power for all three circles.

Formulas

• Radical axis: for S1 = x² + y² + 2g1x + 2f1y + c1 = 0 and S2 = x² + y² + 2g2x + 2f2y + c2 = 0, the radical axis is 2(g1 − g2)x + 2(f1 − f2)y + (c1 − c2) = 0 (valid when the x², y² coefficients match).
• Equal tangent lengths: for P on the radical axis, PT1 = PT2 = √(power of P), since tangent length squared equals power.

Worked Example

Find the radical axis of S1: x² + y² − 4x − 2y − 4 = 0 and S2: x² + y² + 2x − 2y − 11 = 0.

1. Both equations already have x² + y² coefficients equal to 1, so subtract directly: (x² + y² − 4x − 2y − 4) − (x² + y² + 2x − 2y − 11) = 0.
2. The x², y² and y terms all cancel, leaving −4x − 2x − 4 + 11 = −6x + 7 = 0.
3. Hence the radical axis is the vertical line x = 7/6.
4. Check: centres (2, 1) and (−1, 1) give a horizontal line of centres, and x = 7/6 is vertical — perpendicular, as required.

Common Misconceptions

• "The radical axis only exists when the circles intersect." Corrected: it exists for any pair of non-concentric circles — a real line between disjoint circles, the common chord when they intersect.
• "You can subtract the circle equations as written, whatever the coefficients." Corrected: S1 − S2 gives a line only when the x² and y² coefficients match — divide through first if needed.
• "The radical axis passes through the midpoint of the centres." Corrected: it is perpendicular to the line of centres, but only for equal circles is it the perpendicular bisector.

Real-World Applications

• Geometric constructions: the radical axis gives a straightedge-and-compass route to draw common tangents and to locate points with equal tangent lengths to two circles.
• Apollonius-type problems: finding a circle tangent to given circles uses radical centres to reduce the construction to simpler steps.
• Computer graphics: power diagrams partition the plane using radical axes, used in facility location and mesh generation.

In short, the radical axis is the line S1 − S2 = 0 of points with equal power for two circles, perpendicular to the line of centres. Normalise before subtracting; for intersecting circles it is the common chord.