In Geometry, the objects are said to be concentric, when they share the common center. Circles, spheres, regular polyhedra, regular polygons are concentric as they share the same center point. In Euclidean Geometry, two circles that are concentric should have different radii from each other.
The circles with a common center point are known as concentric circles. In other words. It is defined as two or more circles that have the same center point. The region between two concentric circles are of different radii is known as an annulus.
Let the equation of the circle with centre (-g, -f) and radius √[g2+f2-c] be
x2 + y2 + 2gx + 2fy + c =0
Therefore, the equation of the circle concentric with the other circle be
x2 + y2 + 2gx + 2fy + c’ =0
It is observed that both the equations have the same centre (-g, -f), but they have different radii, where c≠ c’
Similarly, a circle with centre (h, k), and the radius is equal to r, then the equation becomes
( x – h )2 + ( y – k )2 = r2
Therefore, the equation of a circle concentric with the circle is
( x – h )2 + ( y – k )2 = r12
Where r ≠ r1
By assigning different values to the radius in the above equation, we shall get a family of circles.
Question: Find the equation of the circle concentric with the circle x2 + y2 + 4x – 8y – 6 =0, having the radius double of its radius.
Given, circle equation: x2 + y2 + 4x – 8y – 6 =0
We know that the equation of the circle is x2 + y2 + 2gx + 2fy + c =0
From the given equation, the center point is (-2, 4)
Therefore, the radius of the given equation will be
r = √[g2+f2-c]
r = √[4+16+6]
r = √26
Let R be the radius of the concentric circle.
It is given that, the radius of the concentric circle is double of its radius, then
R = 2r
R = 2√26
Therefore, the equation of the concentric circle with the radius R and the center point (-g, -f ) is
( x – g )2 + ( y – f )2 = R2
(x + 2)2 + ( y – 4 )2 = (2√26 )2
x2 + 4x + 4 + y2 – 8y + 16 = 4 (26)
x2 + y2 + 4x – 8y +20 = 104
x2 + y2 + 4x – 8y – 84 = 0