Coaxial — Definition, Formula & Examples

Coaxial means sharing the same axis. Two or more geometric figures are coaxial when they are arranged symmetrically around a single common line.

A set of geometric objects (such as circles, cylinders, or cones) is said to be coaxial if there exists a single line—called the common axis or radical axis—with respect to which each object in the set has a symmetric relationship. In the classical sense, a coaxial system of circles consists of all circles in a plane whose pairwise radical axes coincide in one line.

How It Works

To determine whether objects are coaxial, identify whether they share one specific axis of symmetry. For circles in a plane, a coaxial family means every pair of circles in the family has the same radical axis. For three-dimensional objects like cylinders or cones, coaxial simply means they share the same central axis line. A stack of rings on a pole is a physical example: the pole is the common axis, making the rings coaxial.

Worked Example

Problem: Two circles are given:

C_1: x^2 + y^2 = 9

and

C_2: x^2 + y^2 - 6x = 0

. Find the equation of their radical axis and confirm they belong to a coaxial system.

Write in standard form: Rewrite

C_2

by completing the square.

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

Find the radical axis: Subtract the equation of

C_1

from

C_2

. Using the expanded forms

x^2+y^2-9=0

and

x^2+y^2-6x-0=0

(x^2+y^2-6x) - (x^2+y^2-9) = 0 \implies -6x + 9 = 0 \implies x = \tfrac{3}{2}

Interpret: The radical axis is the vertical line

x = \frac{3}{2}

. Any circle that forms this same radical axis with both

C_1

and

C_2

belongs to the same coaxial system.

Answer: The radical axis is

x = \frac{3}{2}

. Since both circles share this radical axis, they form a coaxial pair.

Why It Matters

Coaxial systems of circles appear in coordinate geometry problems on advanced high-school and competition exams. Understanding the radical axis helps you solve problems about intersecting circles, loci, and power of a point efficiently.

Common Mistakes

Mistake: Confusing coaxial with concentric. Students assume both terms mean the same thing.

Correction: Concentric means sharing the same center point. Coaxial means sharing the same axis (a line). Concentric circles all have one center; coaxial circles share a radical axis but typically have different centers.