Circle Equation: Find It With Radius & Center!

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Hey guys! Let's dive into the world of circles and their equations. Today, we're tackling a fun problem: figuring out the equation of a circle given its radius and the center, which happens to be the same as the center of another circle. Sounds like a puzzle, right? Let’s break it down step by step. We'll not only solve this particular problem but also equip you with the knowledge to handle similar questions with confidence. So, grab your thinking caps, and let's get started!

Understanding the Basics of Circle Equations

Before we jump into the problem, let's quickly recap the standard form of a circle equation. This is our trusty tool for this task. The equation looks like this:

(x - h)² + (y - k)² = r²

Where:

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

This equation tells us everything we need to know about a circle: its position in the coordinate plane (center) and its size (radius). Remember this, because it's the key to solving our problem.

Decoding the Given Information

Now, let's look at what we already know. Our target circle has a radius of 2 units. That's a good start! So, we know that r = 2, and therefore r² = 4. This means the right side of our equation will be 4. But what about the center? This is where it gets a bit more interesting. We're told that our circle shares its center with another circle, whose equation is:

x² + y² - 8x - 6y + 24 = 0

This equation isn't in the standard form we love, but don't worry! We can transform it. The process of converting this general form equation into the standard form involves a technique called completing the square. Completing the square allows us to rewrite quadratic expressions in a way that reveals the center coordinates (h, k). It might sound intimidating, but we'll go through it slowly and make it crystal clear. Think of it as a mathematical makeover for our circle equation!

Completing the Square: Finding the Center

Okay, guys, let's roll up our sleeves and complete the square! This technique is super useful for dealing with quadratic equations, and it's exactly what we need here. Remember our equation:

x² + y² - 8x - 6y + 24 = 0

The first step is to group the 'x' terms and the 'y' terms together and move the constant term to the right side of the equation. This helps us focus on each variable separately:

(x² - 8x) + (y² - 6y) = -24

Now comes the fun part: completing the square for both the 'x' and 'y' expressions. To complete the square for an expression like x² + bx, we take half of the coefficient of 'x' (which is 'b'), square it, and add it to both sides of the equation. We do the same for the 'y' terms.

For the 'x' terms:

  • The coefficient of 'x' is -8.
  • Half of -8 is -4.
  • (-4)² = 16

So, we add 16 to both sides of the equation.

For the 'y' terms:

  • The coefficient of 'y' is -6.
  • Half of -6 is -3.
  • (-3)² = 9

So, we add 9 to both sides of the equation.

Our equation now looks like this:

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

Notice that we've added the same values to both sides, maintaining the balance of the equation. Now, the expressions in the parentheses are perfect square trinomials, which means they can be factored into squared binomials:

(x - 4)² + (y - 3)² = 1

Bingo! We've transformed the equation into standard form. Now we can clearly see that the center of this circle is (4, 3). And guess what? This is also the center of the circle we're trying to find the equation for!

Putting It All Together: The Final Equation

Alright, we've got all the pieces of the puzzle. We know:

  • The center of our circle is (4, 3), so h = 4 and k = 3.
  • The radius of our circle is 2, so r = 2 and r² = 4.

Now we just plug these values into the standard form of the circle equation:

(x - h)² + (y - k)² = r²

(x - 4)² + (y - 3)² = 4

(x - 4)² + (y - 3)² = 2²

And there you have it! This is the equation of the circle we were looking for. It has a radius of 2 units and shares its center with the circle defined by the original equation.

Analyzing the Answer Choices

Now, let's take a look at the answer choices provided in the original problem and see which one matches our solution:

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

As we can clearly see, option C. (x - 4)² + (y - 3)² = 2² is the correct answer. It perfectly matches the equation we derived.

Why Other Options Are Incorrect

It's always a good idea to understand why the other options are wrong. This helps solidify your understanding of the concepts.

  • Option A and D: These options have (x + 4) and (y + 3), which would indicate a center at (-4, -3), not (4, 3). Remember, the standard form uses (x - h) and (y - k), so a positive sign inside the parentheses means the coordinate is negative, and vice-versa.
  • Option B: This option has the correct center (4, 3), but the right side of the equation is 2, which would mean the radius is √2, not 2. We need r² on the right side, so it should be 2² = 4.

Key Takeaways and Tips for Success

Awesome job, guys! We've successfully navigated this circle equation problem. Before we wrap up, let's highlight some key takeaways and tips that will help you ace similar problems in the future:

  1. Master the Standard Form: The standard form of a circle equation (x - h)² + (y - k)² = r² is your best friend. Memorize it, understand it, and love it!
  2. Completing the Square is Crucial: This technique is essential for converting general form equations into standard form. Practice it until it becomes second nature. There are tons of online resources and videos that can help you with this.
  3. Pay Attention to Signs: Be careful with the signs when identifying the center coordinates. Remember, (x - h) means the x-coordinate of the center is h, and (y - k) means the y-coordinate of the center is k.
  4. Don't Forget to Square the Radius: The right side of the equation is r², not r. Make sure you square the radius when writing the equation.
  5. Visualize the Circle: If you're struggling, try sketching a quick graph of the circle. This can help you visualize the center and radius and make sure your equation makes sense.

Practice Makes Perfect

Like any skill, mastering circle equations takes practice. The more problems you solve, the more comfortable you'll become with the concepts and techniques. So, seek out practice problems, work through them step-by-step, and don't be afraid to make mistakes. Mistakes are learning opportunities!

Conclusion

So, there you have it! We've successfully found the equation of a circle given its radius and a shared center with another circle. We've reviewed the standard form of a circle equation, mastered the art of completing the square, and learned how to avoid common pitfalls. With these tools in your arsenal, you're well-equipped to tackle any circle equation problem that comes your way. Keep practicing, stay curious, and remember, math can be fun! Until next time, keep circling around with those equations!