Finding The Equation Of A Circle With Radius 2 And Center Determination

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In this comprehensive guide, we will walk you through the process of finding the equation of a circle given its radius and center. This problem often appears in mathematics, particularly in the study of analytic geometry. Understanding how to solve these problems is crucial for students and anyone interested in mathematical applications. We'll explore the concepts, break down the steps, and provide a clear solution. Let's dive in!

Understanding the Basics of Circle Equations

Before we tackle the specific problem, let's establish the fundamental concepts of circle equations. A circle can be defined by its center and radius. The general equation of a circle in the Cartesian coordinate system is given by:

(x – h)² + (y – k)² = r²

Where:

  • (h, k) represents the coordinates of the center of the circle.
  • r represents the radius of the circle.

This equation is derived from the Pythagorean theorem, considering the distance from any point (x, y) on the circle to the center (h, k) is constant and equal to the radius r. Understanding this standard form is essential for manipulating and solving circle-related problems.

Importance of Center and Radius

The center and radius are the two key parameters that define a circle uniquely. The center pinpoints the circle's location in the coordinate plane, while the radius determines its size. When given the center and radius, we can directly plug these values into the standard equation to find the circle's algebraic representation. Conversely, if we are given the equation of a circle, we can determine its center and radius by transforming the equation into the standard form.

Why Understanding Circle Equations Matters

The understanding of circle equations extends beyond theoretical mathematics. It has practical applications in various fields, including:

  • Computer Graphics: Circles are fundamental shapes in computer graphics, used in drawing everything from wheels to complex curves.
  • Physics: The motion of objects in circular paths, such as satellites orbiting the Earth, can be described using circle equations.
  • Engineering: Designing circular structures, like tunnels or pipes, requires a thorough understanding of circle geometry.
  • Navigation: Circles are used in creating maps and determining distances.

By grasping the concepts behind circle equations, you are not just learning math; you are equipping yourself with a tool that has wide-ranging utility.

Problem Statement: Decoding the Circle's Properties

Let's revisit the problem we aim to solve. We are tasked with finding the equation of a circle, given the following information:

  1. The radius of the circle is 2 units.
  2. The center of the circle is the same as the center of another circle whose equation is: x² + y² – 8x – 6y + 24 = 0.

This problem requires us to first identify the center of the given circle equation and then use this center along with the provided radius to construct the equation of the circle we are interested in. This involves a multi-step process, combining algebraic manipulation with geometric understanding.

Breaking Down the Problem

To solve this problem effectively, we can break it down into the following steps:

  1. Find the Center: Determine the center of the circle whose equation is given as x² + y² – 8x – 6y + 24 = 0. This involves completing the square to transform the equation into the standard form.
  2. Use the Radius: Once we have the center, we will use the given radius (2 units) and the center coordinates to write the equation of the desired circle.
  3. Match the Equation: Finally, we will compare our derived equation with the provided options to identify the correct answer.

Why This Approach Works

This systematic approach ensures that we address each aspect of the problem in a logical order. By first finding the center, we establish a crucial piece of information needed to define the new circle. Then, using the radius, we can construct the equation that satisfies the given conditions. This methodical approach minimizes the chances of error and makes the solution process clear.

Step 1: Finding the Center by Completing the Square

The most critical step in solving this problem is to find the center of the circle whose equation is x² + y² – 8x – 6y + 24 = 0. To do this, we will use the method of completing the square. Completing the square is a technique used to rewrite quadratic expressions in a form that reveals the center and radius of the circle more clearly.

Understanding Completing the Square

The process of completing the square involves transforming a quadratic expression (in this case, in both x and y) into a perfect square trinomial plus a constant. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. For example, x² + 2ax + a² is a perfect square trinomial because it can be written as (x + a)². This technique allows us to rewrite the circle equation in the standard form (x – h)² + (y – k)² = r².

Steps to Complete the Square

Let's apply this technique to our equation:

x² + y² – 8x – 6y + 24 = 0

  1. Group x and y terms: Rearrange the equation to group the x terms and y terms together:

    (x² – 8x) + (y² – 6y) + 24 = 0

  2. Complete the square for x: To complete the square for the x terms, take half of the coefficient of x (-8), square it, and add it inside the parenthesis. Half of -8 is -4, and (-4)² is 16. So, we add 16:

    (x² – 8x + 16)

  3. Complete the square for y: Similarly, for the y terms, take half of the coefficient of y (-6), square it, and add it inside the parenthesis. Half of -6 is -3, and (-3)² is 9. So, we add 9:

    (y² – 6y + 9)

  4. Balance the equation: Since we added 16 and 9 inside the parentheses, we must subtract these values from the same side of the equation to maintain balance:

    (x² – 8x + 16) + (y² – 6y + 9) + 24 – 16 – 9 = 0

  5. Rewrite as squared binomials: Now, rewrite the quadratic expressions as squared binomials:

    (x – 4)² + (y – 3)² + 24 – 16 – 9 = 0

  6. Simplify: Simplify the equation by combining the constants:

    (x – 4)² + (y – 3)² – 1 = 0

  7. Isolate the constant: Move the constant to the other side of the equation:

    (x – 4)² + (y – 3)² = 1

Identifying the Center

Now that the equation is in the standard form (x – h)² + (y – k)² = r², we can easily identify the center. Comparing our equation (x – 4)² + (y – 3)² = 1 with the standard form, we see that:

  • h = 4
  • k = 3

Therefore, the center of the circle is (4, 3). This is a crucial piece of information that we will use in the next step.

Step 2: Constructing the Desired Circle Equation

Now that we have found the center of the circle, which is (4, 3), and we know the radius of the new circle is 2 units, we can construct the equation of the desired circle. We will use the standard form of the circle equation:

(x – h)² + (y – k)² = r²

Plugging in the Values

We know:

  • h = 4
  • k = 3
  • r = 2

Substitute these values into the standard equation:

(x – 4)² + (y – 3)² = 2²

Simplifying the Equation

Simplify the equation by squaring the radius:

(x – 4)² + (y – 3)² = 4

This is the equation of the circle with a center at (4, 3) and a radius of 2 units. Now, we can compare this equation with the options provided to find the correct answer.

Step 3: Matching the Equation with the Options

We have derived the equation of the circle as (x – 4)² + (y – 3)² = 4. Now, let's compare this with the given options to find the match:

A. (x + 4)² + (y + 3)² = 2 B. (x – 4)² + (y – 3)² = 2 C. (x – 4)² + (y – 3)² = 2²

Analyzing the Options

  • Option A has the wrong signs for the center coordinates and an incorrect radius.
  • Option B has the correct center coordinates but an incorrect radius.
  • Option C matches our derived equation perfectly. 2² is equal to 4, so the radius is correctly represented.

The Correct Answer

Therefore, the correct equation that represents the circle described is:

C. (x – 4)² + (y – 3)² = 2²

This equation confirms that the center of the circle is at (4, 3) and the radius is 2 units, as specified in the problem.

Conclusion: Mastering Circle Equations

In this guide, we have successfully found the equation of a circle given its radius and a condition on its center. We began by understanding the standard form of a circle equation and the significance of the center and radius. Then, we broke down the problem into manageable steps:

  1. Finding the center of the circle by completing the square.
  2. Constructing the desired circle equation using the center and radius.
  3. Matching the derived equation with the provided options.

By mastering these steps, you can confidently solve similar problems involving circle equations. Remember, the key is to understand the underlying concepts and apply them systematically. Understanding circle equations is not just about solving mathematical problems; it's about developing a logical and methodical approach to problem-solving that can be applied in various real-world scenarios.

Final Thoughts

Circle equations are a fundamental part of analytic geometry and have applications in numerous fields. By practicing and understanding these concepts, you enhance your problem-solving skills and gain a deeper appreciation for the beauty and utility of mathematics. Keep exploring, keep learning, and keep applying these concepts to new challenges!