Coaxial Circles: Explained with Examples, Notes and FAQs

Coaxial circles are a family of circles sharing the same radical axis — the line (real or not) such that every circle in the family has equal power with respect to any point on it. The standard coaxial family is written S + λL = 0, where S = 0 is one circle of the family, L = 0 is the common radical axis line, and λ is a real parameter sweeping through all the circles.

This matters because two intersecting circles define a whole family through their common chord, and problems about "the circle through the intersection of two circles satisfying some condition" are really one-parameter coaxial problems. The family also introduces limiting points — degenerate point-circles of the family — which appear in advanced geometry and in the theory of inversion.

Key Concepts

• Family form: S + λL = 0, i.e. (x² + y² + 2gx + 2fy + c) + λ(ux + vy + w) = 0, still has equal x² and y² coefficients, so it is a circle for every λ.
• Radical axis: for two circles S₁ = 0 and S₂ = 0, the radical axis is S₁ − S₂ = 0, a straight line; it is the common chord when the circles intersect.
• Limiting points: values of λ making the radius zero give point-circles; they lie on the line of centres, on opposite sides of the radical axis.
• Centres: all centres of a coaxial family are collinear on the line of centres, perpendicular to the radical axis.

Formulas

• Coaxial family: x² + y² + 2gx + 2fy + c + λ(ux + vy + w) = 0.
• Centre of the λ-member: (−g − λu/2, −f − λv/2); radius² = (g + λu/2)² + (f + λv/2)² − (c + λw).
• Radical axis of S₁, S₂: 2(g₁ − g₂)x + 2(f₁ − f₂)y + (c₁ − c₂) = 0.
• Units: λ carries 1/length units so that λL has area units like every other term.

Worked Example

Find the limiting points of the coaxial family x² + y² − 4 + λ(x − 6) = 0.

1. Identify the circle part S: x² + y² − 4 = 0 (centre (0, 0), radius 2) and the radical axis L: x − 6 = 0.
2. For the λ-member, 2g = λ so g = λ/2, f = 0, c = −4 − 6λ.
3. Radius squared: r² = g² + f² − c = λ²/4 + 4 + 6λ.
4. Set r² = 0: λ²/4 + 6λ + 4 = 0, i.e. λ² + 24λ + 16 = 0, giving λ = −12 ± 8√2.
5. Centre = (−λ/2, 0), so the limiting points are (6 − 4√2, 0) ≈ (0.34, 0) and (6 + 4√2, 0) ≈ (11.66, 0).
6. Verify the first: with λ = −12 + 8√2, x = 6 − 4√2, y = 0: x² = 68 − 48√2, so x² − 4 + λ(x − 6) = 64 − 48√2 + (−12 + 8√2)(−4√2) = 64 − 48√2 + 48√2 − 64 = 0. Correct — a point-circle.

Common Misconceptions

• "All circles in the family intersect each other." Members of a non-intersecting coaxial family need not meet at all — they share the radical axis, not points.
• "Limiting points lie on the radical axis." They lie on the line of centres; only in special cases do they touch the radical axis.
• "S₁ + λS₂ = 0 is the coaxial form." That is the family through two circles' intersections; the coaxial form S + λL = 0 uses a line, which also covers non-intersecting cases.

Real-World Applications

• Electromagnetics: equipotential circles of two line charges form a coaxial family, with limiting points at the charges.
• Fluid dynamics: streamlines of a source–sink pair are coaxial circles (Apollonius circles).
• Optics: wavefronts refracted through coaxial spherical surfaces are analysed with coaxial-circle geometry.
• GPS trilateration: range circles from collinear stations form a coaxial family whose radical axis locates the receiver's line of position.

In short: S + λL = 0 generates every circle sharing one radical axis, with centres marching along a perpendicular line. Set the radius to zero to find the limiting points, verify them by substitution, and use the radical axis S₁ − S₂ = 0 to build the family from any two circles.