Apollonius Circle — Definition, Formula & Examples

An Apollonius circle is the set of all points whose distances to two fixed points have a constant ratio. If points

A

and

B

are fixed and

k

is a positive constant (with

k \neq 1

), the Apollonius circle is every point

P

where

\frac{PA}{PB} = k

Given two distinct points $A$ and $B$ in the plane and a positive real number $k \neq 1$ , the Apollonius circle with respect to $A$ , $B$ , and $k$ is the locus $\{P : PA/PB = k\}$ . This locus is a circle whose center lies on line $AB$ , and whose diameter endpoints divide segment $AB$ internally and externally in the ratio $k : 1$ .

Key Formula

\frac{PA}{PB} = k, \quad k > 0,\; k \neq 1

Where:

$P$ = Any point on the Apollonius circle
$A$ = First fixed point
$B$ = Second fixed point
$k$ = Constant positive ratio of distances (not equal to 1)

How It Works

To construct an Apollonius circle, start with two fixed points

A

and

B

and choose a ratio

k > 0

k \neq 1

. Find the two points on line

AB

that divide the segment in the ratio

k:1

— one internally and one externally. These two points are the endpoints of a diameter of the Apollonius circle. The center is the midpoint of that diameter. When

k = 1

, the locus degenerates into the perpendicular bisector of

AB

rather than a circle.

Worked Example

Problem: Find the Apollonius circle for points

A = (0, 0)

and

B = (6, 0)

with ratio

k = 2

(meaning

PA/PB = 2

Find the internal division point: Divide

AB

internally in the ratio

2:1

. This point lies at:

D_1 = \frac{1 \cdot A + 2 \cdot B}{2 + 1} = \frac{(0,0) + (12,0)}{3} = (4, 0)

Find the external division point: Divide

AB

externally in the ratio

2:1

. This point lies at:

D_2 = \frac{-1 \cdot A + 2 \cdot B}{2 - 1} = \frac{(0,0) + (12,0)}{1} = (12, 0)

Determine center and radius:

D_1

and

D_2

are endpoints of the diameter. The center is their midpoint and the radius is half the diameter length.

\text{Center} = \left(\frac{4+12}{2},\, 0\right) = (8, 0), \quad r = \frac{12 - 4}{2} = 4

Answer: The Apollonius circle is

(x - 8)^2 + y^2 = 16

, centered at

(8, 0)

with radius

4

Why It Matters

Apollonius circles appear in competition geometry, antenna signal-strength modeling, and any situation where you need the locus of points at a fixed distance ratio from two sources. In coordinate geometry courses, they provide a strong example of how a distance condition translates into the equation of a circle.

Common Mistakes

Mistake: Assuming the ratio

k = 1

also produces a circle.

Correction: When

k = 1

, every point equidistant from

A

and

B

lies on the perpendicular bisector of

AB

, which is a line, not a circle.