Geometry • Topic 19

Radical Axis

The radical axis is the locus of points having equal power with respect to two circles. It is a straight line perpendicular to the line connecting the centers.

Definition

Definition (Radical Axis)

For two non-concentric circles

ω_{1}

and

ω_{2}

, the radical axis is the set of points

P

such that:

P_{ω_{1}} (P) = P_{ω_{2}} (P)

O_{1} P^{2} - R_{1}^{2} = O_{2} P^{2} - R_{2}^{2}

Geometric Properties

Perpendicularity: The radical axis is perpendicular to the line $O_{1} O_{2}$ .
Intersection: If the circles intersect at $A$ and $B$ , the radical axis is the unique line passing through $A$ and $B$ (the common chord).
Tangency: If the circles are tangent at $T$ , the radical axis is the common tangent at $T$ .
No Intersection: If one circle is inside the other (no common points), the radical axis lies outside both.

Construction

To find the radical axis of two non-intersecting circles:

Draw a third auxiliary circle that intersects both.
Draw the two common chords.
Their intersection point $X$ lies on the radical axis.
Drop a perpendicular from $X$ to $O_{1} O_{2}$ .

Practice Problems

Exercise (Problem 1)

Given two circles with radii 3 and 5, and distance between centers 10. Find the distance from the center of the larger circle to the radical axis.

Exercise (Problem 2)

Prove that the common chords of three intersecting circles are concurrent (Radical Center Preview).

Exercise (Problem 3)

Find the equation of the radical axis for circles

x^{2} + y^{2} = 4

and

(x - 3)^{2} + y^{2} = 9