Concentric circles are a fascinating geometric concept. Here’s what you need to know:

Concentric Circles: Definition and Properties

Definition: Concentric circles are a set of circles that share the same center point but have different radii. In other words, they appear nested inside one another. Imagine a dartboard or a ripple of waves in a pond when you throw a stone – those are practical examples of concentric circles.

Equation of Concentric Circles:

Suppose we have a circle with center ((h, k)) and radius (r). Its equation is given by: [ (x – h)^2 + (y – k)^2 = r^2 ]
The key here is that concentric circles have the same center but varying radii.

Region Between Concentric Circles:

The region enclosed by two concentric circles is called an annulus.
The annulus lies between the circles with different radii.

To find the area of the annulus, subtract the area of the smaller circle from the area of the larger circle: [ \text{Area of annulus} = \pi R^2 – \pi r^2 ] where (R) represents the radius of the larger circle, and (r) represents the radius of the smaller circle.

Theorem on Concentric Circles

In two concentric circles, if a chord of the larger circle touches the smaller circle, it is bisected at the point of contact. Let’s break it down:

Given:
Consider two concentric circles (C_1) and (C_2) with center (O).
Let (AB) be a chord of the larger circle (C_1), touching the smaller circle (C_2) at point (P).

Construction:

Join (OP).
To Prove:
(AP = BP)

Proof:

Since (AB) is a chord of the larger circle (C_1), it becomes a tangent to (C_2) at point (P).
(OP) is the radius of circle (C_2).
By the property that the radius is perpendicular to the tangent at the point of contact, we have (OP \perp AB).
Therefore, (OP) bisects the chord (AB), implying that (AP = BP).

Remember, concentric circles are not only mathematically intriguing but also find practical applications in various fields! 🌟

Q1: Concentric Circles and Their Equations

Consider two concentric circles with the same center. The equation of the larger circle is given as: [ x^2 + y^2 + 4x – 8y – 6 = 0 ]

Find the equation of the circle concentric with the given circle, having a radius double that of the larger circle.

Solution:
The given circle has the equation: (x^2 + y^2 + 4x – 8y – 6 = 0).
To find the concentric circle with double the radius, we multiply the coefficients of (x) and (y) by 2: [ (x^2 + 2x) + (y^2 – 4y) + 27 = 0 ]
Completing the square for both (x) and (y): [ (x + 1)^2 + (y – 2)^2 = 32 ]
Therefore, the equation of the concentric circle is: ((x + 1)^2 + (y – 2)^2 = 32).

Q2: Annulus Area Calculation

Given two concentric circles with radii of 8 cm and 12 cm:

Find the length of a chord that is tangent to the smaller circle and passes through the center of the larger circle.

Solution:
The chord is the diameter of the smaller circle.
The diameter of the smaller circle is equal to its radius (8 cm).
Therefore, the length of the chord is 8 cm.