Family of Circles

Two circles in a plane can meet in zero, one, or two points. When they meet in two points, there is a natural question: what other circles also pass through those same two intersection points?

Take the circles x^2 + y^2 - 4x - 6y + 9 = 0 and x^2 + y^2 - 2x - 4y + 1 = 0. Each is a circle. They intersect at two points. And the expression

for different values of \lambda, gives you a different curve through those two points every time. When \lambda = 0, you get the first circle back. When \lambda = 1, you get a new curve. When \lambda = -1, you get something that looks different again.

Why does every member of this family pass through the intersection points? The argument is identical to the one for families of lines. If a point P lies on both circles, then S_1(P) = 0 and S_2(P) = 0. So S_1(P) + \lambda S_2(P) = 0 + \lambda \cdot 0 = 0, regardless of \lambda. The intersection points satisfy the family equation automatically, for every \lambda.

Circles through the intersection of two circles

The general family

Let S_1 \equiv x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0 and S_2 \equiv x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0 be two circles.

Expand S_1 + \lambda S_2 = 0:

Divide by (1 + \lambda) (valid when \lambda \neq -1) to put it in standard circle form:

The centre of this circle is \left(-\dfrac{g_1 + \lambda g_2}{1 + \lambda},\; -\dfrac{f_1 + \lambda f_2}{1 + \lambda}\right). As \lambda varies, this centre traces out a straight line — the line joining the centres of S_1 and S_2. The family is a one-parameter collection of circles whose centres all lie on the same line, and which all pass through the same two points.

The special case \lambda = -1: the radical axis

When \lambda = -1, the equation S_1 + \lambda S_2 = S_1 - S_2 = 0 reduces to:

The x^2 and y^2 terms cancel. What remains is a linear equation — a straight line, not a circle. This line is the radical axis of the two circles, and it is one of the most important constructions in circle geometry.

Radical axis

What it is geometrically

The radical axis is the locus of all points that have equal power with respect to both circles.

Recall from the article on tangents that the power of a point P(x_1, y_1) with respect to a circle S = 0 is S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c. When P is outside the circle, \sqrt{S_1} is the length of the tangent from P to the circle. The power is the square of the tangent length.

A point P lies on the radical axis of circles S_1 = 0 and S_2 = 0 when its power with respect to both circles is the same. That means the tangent drawn from P to S_1 has the same length as the tangent drawn from P to S_2.

The derivation

Set the powers equal. For a point (x, y):

The x^2 + y^2 terms cancel:

Why it is perpendicular to the line of centres

The line of centres has the direction (g_1 - g_2, f_1 - f_2) (since the centres are (-g_1, -f_1) and (-g_2, -f_2)). The radical axis 2(g_1 - g_2)x + 2(f_1 - f_2)y + (c_1 - c_2) = 0 has normal direction (g_1 - g_2, f_1 - f_2). For a line ax + by + c = 0, the direction along the line is (b, -a), and the normal direction is (a, b). The normal to the radical axis is exactly the direction of the line of centres. So the radical axis is perpendicular to the line of centres.

When the circles intersect

If the two circles actually intersect at two points A and B, then both A and B have zero power with respect to both circles — they lie on both. So the radical axis passes through A and B. In this case, the radical axis is simply the common chord of the two circles.

When the circles do not intersect, the radical axis still exists as a line between them — but it does not cross either circle. A point on this line is equidistant (in the tangent-length sense) from both circles, even though it cannot touch either.

A worked computation

Take S_1 \equiv x^2 + y^2 - 4x - 4y + 4 = 0 (centre (2, 2), radius 2) and S_2 \equiv x^2 + y^2 - 12x - 4y + 24 = 0 (centre (6, 2), radius 4).

S_1 - S_2 = (-4x + 12x) + (-4y + 4y) + (4 - 24) = 8x - 20 = 0, which gives x = 5/2.

The radical axis is the vertical line x = 2.5. Check: the perpendicular from C_1(2, 2) to x = 2.5 has length 0.5, and from C_2(6, 2) it has length 3.5. The tangent lengths from a point on x = 2.5, say (2.5, 2), are: to S_1: \sqrt{6.25 + 4 - 10 - 8 + 4} = \sqrt{-3.75} — this point is inside S_1. That is fine: the power is -3.75 for both circles. Check S_2 at (2.5, 2): 6.25 + 4 - 30 - 8 + 24 = -3.75. Equal powers, as promised.

Radical centre

Three circles, three radical axes

Given three circles S_1, S_2, S_3, you can form three radical axes:

• Radical axis of S_1 and S_2: S_1 - S_2 = 0
• Radical axis of S_2 and S_3: S_2 - S_3 = 0
• Radical axis of S_1 and S_3: S_1 - S_3 = 0

These three lines are concurrent — they all pass through one point. That point is called the radical centre.

Why are the three radical axes concurrent? Add the first two equations: (S_1 - S_2) + (S_2 - S_3) = S_1 - S_3. So any point satisfying the first two radical axes automatically satisfies the third. The three lines cannot be independent — the third is a consequence of the first two. They must share a common point.

The radical centre has a beautiful geometric property: from this single point, the tangent drawn to any of the three circles has the same length. If the radical centre lies outside all three circles, you can draw a circle centred at the radical centre with this common tangent length as its radius. This circle cuts all three given circles at right angles — which leads directly to the notion of orthogonal circles.

Orthogonal circles

Two circles are orthogonal if they intersect at right angles — meaning the tangent to each circle at a point of intersection is perpendicular to the tangent to the other circle at that same point.

Since the tangent to a circle at any point is perpendicular to the radius at that point, the condition "tangents perpendicular at intersection" is the same as "radii perpendicular at intersection." At a point P where the two circles meet, the radius of circle 1 to P is perpendicular to the radius of circle 2 to P.

Deriving the condition

Let the circles be S_1 \equiv x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0 and S_2 \equiv x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0, with centres C_1(-g_1, -f_1) and C_2(-g_2, -f_2) and radii r_1 and r_2.

At a point of intersection P, the segments C_1P and C_2P are the two radii, and they are perpendicular. The triangle C_1PC_2 is a right triangle with the right angle at P. By Pythagoras:

Compute C_1C_2^2:

And r_1^2 + r_2^2 = (g_1^2 + f_1^2 - c_1) + (g_2^2 + f_2^2 - c_2).

Setting C_1C_2^2 = r_1^2 + r_2^2:

Cancel g_1^2 + g_2^2 + f_1^2 + f_2^2 from both sides:

This is a clean algebraic condition — no square roots, no trigonometry, just a linear relation between the coefficients. You can check orthogonality by reading off g, f, c from each circle and substituting.

Verifying with an example

Take S_1 \equiv x^2 + y^2 - 6x - 2y + 6 = 0 (so g_1 = -3, f_1 = -1, c_1 = 6; centre (3, 1), r_1 = 2) and S_2 \equiv x^2 + y^2 - 4y - 2 = 0 (so g_2 = 0, f_2 = -2, c_2 = -2; centre (0, 2), r_2 = \sqrt{6}).

Check: 2g_1g_2 + 2f_1f_2 = 2(-3)(0) + 2(-1)(-2) = 0 + 4 = 4.

And c_1 + c_2 = 6 + (-2) = 4.

Since 2g_1g_2 + 2f_1f_2 = c_1 + c_2, the circles are orthogonal.

Cross-check with Pythagoras: C_1C_2^2 = 9 + 1 = 10. And r_1^2 + r_2^2 = 4 + 6 = 10. Equal, as required.

Worked examples

Common confusions

• "S_1 + \lambda S_2 = 0 always gives a circle." Not when \lambda = -1. For \lambda = -1, the x^2 and y^2 terms cancel and you get a straight line — the radical axis. This is the one "degenerate" member of the family.

• "Two non-intersecting circles have no radical axis." They do. The radical axis exists for any two distinct circles, whether they intersect, are tangent, or are completely separate. When the circles do not intersect, the radical axis falls in the region between them (or outside both), but it is still a well-defined line.

• "The radical axis passes through the centres." No — the radical axis is perpendicular to the line joining the centres. It never passes through either centre (unless both circles have the same radius, in which case it passes through the midpoint of the line of centres, but still not through the centres themselves).

• "Orthogonal circles don't intersect." They must intersect — orthogonality is a condition at the points of intersection. If two circles do not intersect, the concept of orthogonality does not apply (or rather, it extends via the algebraic condition 2g_1g_2 + 2f_1f_2 = c_1 + c_2, which can hold even for non-intersecting circles, but the geometric meaning of "right angle at intersection" requires actual intersection points).

Going deeper

Coaxial families

When you write S_1 + \lambda S_2 = 0 for all \lambda, you get a family of circles that all share the same radical axis (the line S_1 - S_2 = 0). Such a family is called a coaxial system. Every pair of circles in a coaxial system has the same radical axis — which is why the system is called "co-axial" (sharing one axis).

There are three types of coaxial systems:

1. Intersecting type: all circles pass through two common real points (the points where the radical axis meets any circle in the family).
2. Tangent type: all circles pass through one common point (the radical axis is tangent to every circle in the family).
3. Non-intersecting type: no two circles in the family share a common point (the radical axis does not meet any circle in the family).

In each case, the centres of all circles in the family lie on a single line — the line of centres — which is perpendicular to the radical axis.

The orthogonal family

Given a coaxial system, there exists another coaxial system such that every circle in one family is orthogonal to every circle in the other. These two families are called conjugate coaxial systems. The radical axis of one family is the line of centres of the other, and vice versa.

This duality is a beautiful piece of geometry. The intersecting-type system (whose circles share two common points) pairs with a non-intersecting-type system (whose circles all have the same line of centres passing through those two points). The tangent-type system pairs with itself.

Power of a point as a unifying concept

The notion of power of a point is the thread connecting everything in this article:

• The tangent length from an external point is \sqrt{\text{power}}.
• The radical axis is the locus of equal power with respect to two circles.
• The radical centre has equal power with respect to three circles.
• A circle is the locus of points with zero power with respect to itself.
• The chord of contact from a point is determined by the power.

The power of a point with respect to a circle is one number — S_1 — and that one number controls the geometry of tangents, chords, polars, and radical axes. When Bhaskara II studied circle configurations in the Lilavati, he used equivalent distance-based arguments; the algebraic encoding via S_1 came later with the coordinate methods of analytic geometry, but the geometric insight is the same.

Where this leads next

Families of circles and the radical axis are foundational for the more advanced circle theory and for conics:

• Circle — Standard Forms — the equations of circles in standard and general form.
• Family of Lines — the same one-parameter family idea, applied to lines rather than circles.
• Pair of Tangents and Chord of Contact — the tangent pair and chord of contact, closely related to the power of a point.
• Line and Circle — when a line meets a circle, and the conditions for tangency and secancy.
• Locus — the general method for finding equations of geometric loci.