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Concentric Circles Equation
A concentric circle equation is a mathematical equation that defines a circle that has a center point and radii that are all the same length. The equation can be used to define the location of the center point and the size of the circle.
Concentric Circles Meaning
The circles with a common centre are known as concentric circles and have different radii. In other words, it is defined as two or more circles that have the same centre point. The region between two concentric circles are of different radii is known as an annulus.
Concentric Circle Equations
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
Concentric Circles – Theorem
In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
Proof
Given:
Consider two concentric circles C1 and C2, with centre O and a chord AB of the larger circle C1, touching the smaller circle C2 at the point P as shown in the figure below.
Construction:
Join OP.
To prove: AP = BP
Proof:
Since AB is the chord of larger circle C1, it becomes the tangent to C2 at P.
OP is the radius of circle C2.
We know that the radius is perpendicular to the tangent at the point of contact.
So, OP ⊥ AB
Now AB is a chord of the circle C1 and OP ⊥ AB.
Therefore, OP is the bisector of the chord AB.
Thus, the perpendicular from the centre bisects the chord, i.e., AP = BP.
