Finding The Equation Of A Circle Given Its Radius And Center

Jul 16, 2025 by Sam Evans 61 views

Which Equation Represents the Circle Described?

Hey everyone! Today, we're diving into the fascinating world of circles and their equations. We've got a cool problem on our hands that involves finding the equation of a circle given its radius and center, with a little twist – the center is related to another circle's equation. This is a classic problem that combines algebraic manipulation and geometric understanding, so let's break it down step by step. You will understand circle equations better after this article. So, buckle up, grab your thinking caps, and let's get started!

Understanding the Basics of Circle Equations

Before we jump into the problem, let's quickly review the standard form of a circle's equation. The general equation of a circle with center $(h, k)$ and radius $r$ is given by:

$(x - h)^2 + (y - k)^2 = r^2$

This equation tells us a lot. The values $h$ and $k$ give us the coordinates of the center, and $r$ tells us the radius. Our main goal today is to find an equation in this form. We know the radius, so that's one piece of the puzzle. The tricky part is finding the center, which is linked to another circle's equation. We'll use a technique called "completing the square" to unravel the center of the given circle. Stay tuned; it's like detective work with equations!

Step-by-Step Guide to Completing the Square

Completing the square is a powerful technique in algebra that allows us to rewrite quadratic expressions in a more convenient form. In the context of circles, it helps us transform the given equation into the standard form, making it easy to identify the center. Let's walk through the process step-by-step:

Group the x and y terms: We'll start by rearranging the given equation, grouping the $x$ terms together and the $y$ terms together. This will help us focus on each variable separately.
Complete the square for x: Take the coefficient of the $x$ term, divide it by 2, square the result, and add it to both sides of the equation. This creates a perfect square trinomial for the $x$ terms.
Complete the square for y: Do the same for the $y$ terms. Take the coefficient of the $y$ term, divide it by 2, square the result, and add it to both sides. This will give us a perfect square trinomial for the $y$ terms.
Rewrite as squared binomials: Now, we can rewrite the perfect square trinomials as squared binomials. This is where the magic happens, and we start to see the standard form of the circle equation emerge.
Simplify the equation: Combine the constants on the right side of the equation. The equation is now in the standard form $(x - h)^2 + (y - k)^2 = r^2$ , and we can easily identify the center $(h, k)$ and the radius $r$ .

Completing the square might sound intimidating, but it becomes second nature with practice. Remember, it's all about turning those quadratic expressions into perfect squares, making our lives much easier when dealing with circle equations.

Problem Breakdown: Finding the Circle's Center

Let's dive into our specific problem. We are given the equation of a circle:

$x^2 + y^2 - 8x - 6y + 24 = 0$

Our mission, should we choose to accept it (and we do!), is to find the center of this circle. To do this, we'll employ our trusty tool: completing the square. Follow along, guys, and you'll see how this works.

Applying Completing the Square to the Given Equation

First, we'll group the $x$ and $y$ terms:

$(x^2 - 8x) + (y^2 - 6y) + 24 = 0$

Next, let's complete the square for the $x$ terms. We take half of the coefficient of $x$ (-8), which is -4, and square it to get 16. We'll add 16 to both sides of the equation. For the $y$ terms, we take half of the coefficient of $y$ (-6), which is -3, and square it to get 9. We'll also add 9 to both sides:

$(x^2 - 8x + 16) + (y^2 - 6y + 9) + 24 = 16 + 9$

Now, we can rewrite the expressions in parentheses as squared binomials:

$(x - 4)^2 + (y - 3)^2 + 24 = 25$

Finally, we'll subtract 24 from both sides to isolate the squared terms:

$(x - 4)^2 + (y - 3)^2 = 1$

Voilà! We've transformed the equation into the standard form. From this, we can clearly see that the center of the circle is $(4, 3)$ . Great job, team! We've cracked the first part of the puzzle.

Solving the Problem: Constructing the New Circle Equation

Now that we've found the center of the given circle, which is $(4, 3)$ , we know that this is also the center of the new circle we're trying to find. We also know that the radius of our new circle is 2 units. Remember the standard equation of a circle?

$(x - h)^2 + (y - k)^2 = r^2$

Where $(h, k)$ is the center and $r$ is the radius. We've got all the pieces we need! Let's plug in the values:

$(x - 4)^2 + (y - 3)^2 = 2^2$

Simplifying, we get:

$(x - 4)^2 + (y - 3)^2 = 4$

And there we have it! This is the equation of the circle with center $(4, 3)$ and radius 2. Comparing this to the answer choices, we can see that option C, $(x - 4)^2 + (y - 3)^2 = 2^2$ , is the correct answer.

Why Other Options Are Incorrect

It's always a good idea to understand why the other options are wrong. Let's take a quick look:

Option A: $(x + 4)^2 + (y + 3)^2 = 2$ . This equation represents a circle with center $(-4, -3)$ and radius $\sqrt{2}$ , which doesn't match our requirements.
Option B: $(x - 4)^2 + (y - 3)^2 = 2$ . This equation has the correct center $(4, 3)$ , but the radius is $\sqrt{2}$ , not 2.
Option D: This isn't an equation at all, so it can't be the answer.

Understanding why incorrect options are wrong reinforces our understanding of the correct solution and the concepts involved. It's a great way to solidify your knowledge!

Conclusion: Mastering Circle Equations

Wow, we did it! We successfully found the equation of the circle by first determining its center using the method of completing the square, and then applying the standard circle equation. This problem was a fantastic exercise in combining algebraic techniques with geometric concepts. Remember, the key to mastering these types of problems is practice, practice, practice! The more you work with circle equations, the more comfortable and confident you'll become.

Key Takeaways and Tips for Success

Before we wrap up, let's highlight some key takeaways and tips that will help you tackle similar problems in the future:

Master the Standard Circle Equation: Knowing the standard form $(x - h)^2 + (y - k)^2 = r^2$ is crucial. Memorize it, understand it, and love it!
Completing the Square is Your Friend: This technique is invaluable for finding the center and radius of a circle when the equation is not in standard form. Practice it until it becomes second nature.
Pay Attention to Details: Small errors can lead to incorrect answers. Double-check your work, especially when dealing with signs and arithmetic.
Visualize the Problem: Whenever possible, try to visualize the circle and its properties. This can help you catch mistakes and develop a better intuition for the problem.
Practice Makes Perfect: The more problems you solve, the better you'll become. Don't be afraid to tackle challenging problems – they're the best way to learn!

So, guys, keep practicing, keep exploring, and keep those equations coming! You've got this!