Circles are symbolically very important in today’s world. There are various types of circles that we study here. We are going to discuss concentric circles.

Concentric Circles Definition

Circles that have the same center point are known as concentric circles, they fit within each other and have the same distance apart from each way around. Having different radius, the region between two concentric circles is known as the annulus. According to Euclidean Geometry, the condition for two circles being concentric is that they must not have the same radii.

Equations of Concentric Circles

Let us consider the equation of the circle with centre (-g, -f) and radius of the same √[g2+f2-c] is x2 + y2 + 2gx + 2fy + c =0

So, the equation of the Circle that is concentric with the other Circle is

x2 + y2 + 2gx + 2fy + c’ =0

It is very clear that both the equations come with a common center (-g, -f), but they have different radii, where c≠ c’, which is a necessary condition for the circles to be concentric.

In the same way let there be a circle with center (h, k), and the radius is equal to r, then the equation will be

( x – h )2 + ( y – k )2 = r2

And, the equation of a circle which is concentric with another circle will be

( x – h )2 + ( y – k )2 = r12

Where r is not equal to r1

By putting different values to the radius in the above equations, we will supposedly get a family of concentric circles.

Examples of Concentric Circles

Let us understand concentric circles with the help of the following question-answer type example.

Question: Determine the equation of the Circle, which is concentric with the Circle having equation x2 + y2 + 4x – 8y – 6 =0, given that the radius double of that this Circle.

Answer:

We’ve been provided with, circle equation: x2 + y2 + 4x – 8y – 6 =0

And we already know that the equation of the circle is x2 + y2 + 2gx + 2fy + c =0

So, from the given equation, the center point of the circles will be(-2, 4)

So, the radius of the given equation will be as follows.

r = √[g2+f2-c]

r = √[4+16+6]

r = √26

Let us assume R is the radius of the concentric Circle.

We’ve been given that, the radius of the concentric Circle is double that of its original radius, then.

R = 2r

Putting the value in the above equation,

R equals to 2√26

Therefore, the equation of the circle which is concentric and have the radius R and the center point (-g, -f ) is

( x – g )2 + ( y – f )2 equals to R2

(x + 2)2 + ( y – 4 )2 equals to (2√26 )2

Now,

x2 + 4x + 4 + y2 – 8y + 16 equals to 4 (26)

x2 + y2 + 4x – 8y +20 equals to 104

So,

x2 + y2 + 4x – 8y – 84 equals to 0

Hence, derived.

Practical Examples of Concentric Circles

1. When we drop something into water, naturally, it has an expanding system of forming concentric circles, which are called ripples. The center is the point where the object has been dropped.
2. Electrical Cables, which is known as coaxial cable in which the live core is surrounded by neutral and earth core-forming cylindrical shells, is a good example of concentric circles.
3. Another example is Diopter Sights, and they are the type of sights found on the target rifles commonly. When these sights are completely aligned, the point for the impact will be exactly in the middle of the front sight circle.

Difference between concentric and eccentric.

People usually confuse between these two, so to avoid that, we must understand what an eccentric circle is.

The circles which are usually contained within another circle are called eccentric circles, given that they may or may not have the same center. When they share the same center, they can be called as concentric circles.

FAQ