Radical Axis and Coaxal System of Circles
On this page
Direct answer
Subtract two circles' equations (after matching the x^2 + y^2 coefficients) and the quadratic terms cancel, leaving a line: S1 − S2 = 0, the radical axis, the locus of equal powers with respect to the two circles. It is perpendicular to the line joining the centres, and for three circles the three radical axes concur at the radical centre — the single point with equal power to all three, and the centre of the unique circle cutting all three orthogonally. The power of a point against S = x^2 + y^2 + 2gx + 2fy + c is the equation's value S1 at that point, and the tangent length from an external point is √S1.
What you must remember
- Radical axis: S1 − S2 = 0 is a straight line perpendicular to the line of centres; at every point on it, the tangent lengths to the two circles are equal.
- Concentric exception: distinct concentric circles have no radical axis — their equations differ by a nonzero constant, which never vanishes.
- Radical centre: the three radical axes of three circles (with non-collinear centres) meet at one point of equal power to all three.
- Orthogonal circle: centred at the radical centre with radius equal to the tangent length to any of the three given circles.
- Tangent length: from (x1, y1) to circle S = 0 it is √S1; S1 > 0 outside the circle, S1 = 0 on it, S1 < 0 inside (imaginary tangent).
- Coaxal system: circles sharing one common radical axis; the standard form x^2 + y^2 + 2gx + c = 0 with c fixed and g varying has all centres on the x-axis and radical axis x = 0.
- Limiting points: the point-circles of a coaxal system, found by setting its radius √(g^2 − c) to zero — for c > 0 they sit at (±√c, 0) in the standard form.
Locating a radical centre
Take C1: x^2 + y^2 = 4, C2: x^2 + y^2 − 2x = 0 and C3: x^2 + y^2 − 2y = 0. Subtracting C2 from C1 gives 2x − 4 = 0, so x = 2; subtracting C3 from C1 gives 2y − 4 = 0, so y = 2. The radical centre is (2, 2) — and only two subtractions were needed, since the third axis passes through automatically. The power of (2, 2) against C1 is 4 + 4 − 4 = 4, so the tangent length is 2. The circle centred at (2, 2) with radius 2 cuts all three circles orthogonally: its centre sits at the equal-power point, and its radius is exactly the tangent each original circle contributes. This two-subtraction rhythm — normalise, subtract pairwise, read off the intersection — resolves nearly every radical-axis question JEE Main fields, and the orthogonal circle is the usual final sentence of the Advanced versions.
Where marks leak
The opening technicality: before subtracting, both equations must carry coefficient 1 on x^2 and y^2 — a circle written as 2x^2 + 2y^2 + ... must be halved first, or the "line" produced is nonsense. JEE Main asks for a radical axis or an equal-tangent-length comparison; JEE Advanced prefers limiting points of a coaxal system, orthogonal trajectories, and the circle-through-intersection rewritten as a coaxal member. The recurring conceptual losses: expecting a radical axis from concentric circles; reading a negative power as a length (tangent lengths are real only outside or on the circle); and forgetting that the radical centre, not any centre of the original circles, is the natural home of the orthogonal circle. A final habit: verify the found point on one circle not used in the subtraction — one substitution certifies the whole construction.
Frequently asked questions
What is the radical axis of two circles?
The line S1 − S2 = 0, the locus of equal powers, always perpendicular to the line joining the centres.
What is the radical centre of three circles?
The common point of their three radical axes, having equal power with respect to all three circles.
How is the length of the tangent from (x1, y1) computed?
As √S1, where S1 is the circle's equation evaluated at the point.
What defines a coaxal system of circles?
A family of circles sharing one common radical axis, representable as x^2 + y^2 + 2gx + c = 0 with c fixed.
Can concentric circles have a radical axis?
No finite one — their equations differ only by a constant, which is never zero for distinct circles.