Equation of Circle 2 Given Distance from Circle 1 Center

In this article, we explore the equation of Circle 2 when it is located anywhere that is x units away from the center of Circle 1. We will derive the equation of Circle 2, taking into account the given relationships between the radii and centers of the two circles.

Introduction

The given circles are Circle 1 and Circle 2. Circle 1 has a radius of 2x units, whereas Circle 2 has a radius of x units. We aim to find the equation of Circle 2 given that it is located x units away from the center of Circle 1, denoted as (a, b).

Equation of Circle 1

The equation for Circle 1 with center (a, b) and a radius of 2x can be expressed as:

(x - a)2 (y - b)2 4x2

Position of the Center of Circle 2

Let us consider the center of Circle 2 as a point P, which is r units away from the center of Circle 1, (a, b). We can express this point as PP(ar cos θ, br sin θ), where θ is the angle subtended by the line segment from (a, b) to the center of Circle 2.

Equation of Circle 2

Given that the radius of Circle 2 is simply x, we can derive its equation. The equation of a circle with center (h, k) and radius r is given by:

(x - h)2 (y - k)2 r2

Applying these values to our situation where the center of Circle 2 is at (ar cos θ, br sin θ) and the radius is x, we obtain:

(x - ar cos θ)2 (y - br sin θ)2 x2

Final Equation of Circle 2

Let us now simplify and eliminate r2. By substituting 4r2 from the equation of Circle 1 (Equation 1), we can express r in terms of the given parameters and simplify the equation for Circle 2 as follows:

(x - ar cos θ)2 (y - br sin θ)2 x2

Thus, the equation of Circle 2, with its center at a point which is r units away from the center of Circle 1, is given by:

(x - ar cos θ)2 (y - br sin θ)2 x2

Conclusion

We have derived the equation of Circle 2, considering the distance from its center to the center of Circle 1. The process involved using the given radii and centers of the circles and the distance formula. The final equation is useful in understanding how the position and radius of a circle affect its equation, given a specific distance from another circle's center.

Frequently Asked Questions (FAQs)

1. How can I determine the equation of Circle 2 if its center is anywhere that is r units away from the center of Circle 1?

By using the coordinates of the center of Circle 1 and applying the distance formula, we can find the coordinates of the center of Circle 2. These coordinates can then be used to derive the equation of Circle 2, which is given by:

(x - ar cos θ)2 (y - br sin θ)2 x2

2. Can I use this equation to determine the position of Circle 2 in relation to Circle 1?

Yes, by knowing the value of θ (the angle subtended by the line segment from (a, b) to the center of Circle 2), you can determine the exact position of Circle 2 relative to Circle 1. The angle θ can be calculated using the coordinates of the centers of the circles and the given distance.

3. What is the significance of the term 4r2 in the equation of Circle 1?

The term 4r2 in the equation of Circle 1, (x - a)2 (y - b)2 4x2, represents the square of the radius of Circle 1 multiplied by 4. This term helps in determining the radius of Circle 1, and it is essential in the process of finding the equation of Circle 2, as it allows us to express the distance r in terms of the given parameters.