Equation Of A Circle: Center (-2, -4), Passes Through (3, 8)

Hey everyone! Today, we're diving into the fascinating world of circles in coordinate geometry. Specifically, we're going to tackle a common question: how to find the equation of a circle when you know its center and a point it passes through. It might sound intimidating at first, but trust me, with a few key concepts and a bit of algebraic maneuvering, you'll be solving these problems like a pro! Let's break it down. Before we jump into a specific problem, let's quickly recap the standard equation of a circle. This is our fundamental tool, so making sure we understand it is key. The standard form equation for a circle with center (h, k) and radius r is: (x - h)² + (y - k)² = r². Notice that the center coordinates (h, k) are subtracted from x and y, respectively. Also, the radius r is squared in the equation. Keeping these points in mind will help you avoid common mistakes. Now that we've refreshed the basics, let's get our hands dirty with an example. Suppose we need to find the equation of a circle whose center is located at the point (-2, -4) and which passes through the point (3, 8). This is exactly the kind of problem that might show up in your math class or on a standardized test, so let's take it step by step. Our first task is to identify the values of h, k, and r in our standard equation. We already know the center of the circle is (-2, -4). Comparing this to the general form (h, k), we can directly see that h = -2 and k = -4. So far, so good! Now, the tricky part: finding the radius, r. Remember, the radius is the distance from the center of the circle to any point on the circle. We know the center (-2, -4) and a point on the circle (3, 8). So, to find the radius, we need to calculate the distance between these two points. This is where the distance formula comes to our rescue.

Using the Distance Formula to Calculate the Radius

Let's talk about how to use the distance formula to calculate the radius. The distance formula is derived from the Pythagorean theorem and provides a way to find the distance between two points in a coordinate plane. Given two points (x₁, y₁) and (x₂, y₂), the distance between them, d, is given by: d = √((x₂ - x₁)² + (y₂ - y₁)²). This formula might look a bit intimidating, but it's really just a matter of plugging in the correct values and doing some arithmetic. In our case, we want to find the distance between the center of the circle (-2, -4) and the point (3, 8) that the circle passes through. So, we can assign (x₁, y₁) = (-2, -4) and (x₂, y₂) = (3, 8). Now, let's substitute these values into the distance formula: r = √((3 - (-2))² + (8 - (-4))²). Notice that we're substituting carefully, paying attention to the negative signs. This is a common place for errors, so it's always a good idea to double-check your work. Next, we simplify the expressions inside the parentheses: r = √((3 + 2)² + (8 + 4)²). This gives us: r = √(5² + 12²). Now we need to calculate the squares: r = √(25 + 144). And finally, we add the numbers under the square root: r = √169. The square root of 169 is 13, so we find that the radius of our circle is r = 13. Great job! We've successfully calculated the radius. Now that we have both the center (h, k) = (-2, -4) and the radius r = 13, we have all the pieces we need to write the equation of the circle. Remember, the standard form equation of a circle is (x - h)² + (y - k)² = r². Let's substitute the values we found into this equation.

Substituting Values into the Circle Equation

Alright guys, let's substitute the values into the circle equation and see how it all comes together. We know that the standard form equation of a circle is (x - h)² + (y - k)² = r². We've already determined that the center of our circle is (h, k) = (-2, -4) and the radius is r = 13. Now, it's just a matter of carefully plugging these values into the equation. Substituting h = -2 and k = -4 into the equation, we get: (x - (-2))² + (y - (-4))² = 13². Notice how we're being extra careful with the negative signs. Subtracting a negative number is the same as adding a positive number, so we can simplify this to: (x + 2)² + (y + 4)² = 13². Now, we need to calculate 13², which is 13 * 13 = 169. So, our equation becomes: (x + 2)² + (y + 4)² = 169. And there you have it! This is the equation of the circle with center (-2, -4) that passes through the point (3, 8). How cool is that? We've taken a seemingly complex problem and broken it down into manageable steps. Let's just take a moment to appreciate what we've accomplished. We started with the standard equation of a circle, used the distance formula to find the radius, and then substituted everything back into the equation. This is a powerful problem-solving strategy that you can apply to many different math problems. Now, let's compare our result to the answer choices provided in the original question. We were given four options:

A. (x - 2)² + (y + 4)² = 144 B. (x + 2)² + (y + 4)² = 169 C. (x + 2)² + (y + 4)² = 144 D. (x - 2)² + (y + 4)² = 169

Comparing the Result with the Answer Choices

Okay, let's carefully compare our result with the answer choices. We found the equation of the circle to be (x + 2)² + (y + 4)² = 169. Now, we need to see which of the given options matches our result exactly. Option A is (x - 2)² + (y + 4)² = 144. Notice that the x term has (x - 2)², which is different from our (x + 2)². Also, the right-hand side is 144, while ours is 169. So, option A is not the correct answer. Option B is (x + 2)² + (y + 4)² = 169. This looks promising! The left-hand side matches our result perfectly, and the right-hand side is also 169. It seems like we've found our answer, but let's just double-check the other options to be sure. Option C is (x + 2)² + (y + 4)² = 144. The left-hand side matches our result, but the right-hand side is 144, not 169. So, option C is incorrect. Option D is (x - 2)² + (y + 4)² = 169. Again, the x term has (x - 2)², which doesn't match our (x + 2)². So, option D is also incorrect. After carefully comparing our result with all the answer choices, we can confidently conclude that the correct answer is option B: (x + 2)² + (y + 4)² = 169. Fantastic! We've not only found the equation of the circle but also verified our answer against the given options. This is a great way to ensure accuracy, especially in exam situations. So, what are the key takeaways from this problem? First, remember the standard form equation of a circle: (x - h)² + (y - k)² = r². Second, use the distance formula to find the radius when you know the center and a point on the circle. And third, be careful with negative signs and double-check your work at each step. These tips will help you tackle similar problems with confidence. Now, let’s recap the entire process we followed. This will solidify your understanding and make it easier to apply these steps to other circle equation problems.

Recap and Final Thoughts on Circle Equations

Let's recap the entire process and share some final thoughts on circle equations. We started by understanding the standard form equation of a circle: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Then, we identified the center of the circle from the problem statement, which was given as (-2, -4). This gave us the values h = -2 and k = -4. The next crucial step was finding the radius. Since we knew a point on the circle (3, 8) and the center (-2, -4), we used the distance formula: r = √((x₂ - x₁)² + (y₂ - y₁)²) to calculate the distance between these two points. This gave us the radius r = 13. Once we had the center (h, k) and the radius r, we substituted these values into the standard form equation: (x - (-2))² + (y - (-4))² = 13². Simplifying this, we arrived at the equation of the circle: (x + 2)² + (y + 4)² = 169. Finally, we compared our result with the given answer choices and confirmed that option B was the correct answer. Throughout this process, we emphasized the importance of being careful with negative signs and double-checking your work. These small details can make a big difference in getting the correct answer. So, what's the big picture here? Finding the equation of a circle might seem like a specific problem, but it illustrates some important general problem-solving strategies in mathematics. Breaking down a complex problem into smaller, manageable steps is a key skill. Also, understanding and applying fundamental formulas like the standard form equation of a circle and the distance formula are essential. And finally, careful attention to detail and double-checking your work are always good habits to cultivate. So, the next time you encounter a problem involving circles, remember these steps and strategies. You've got this! And remember, math is not just about finding the right answer; it's about understanding the process and developing your problem-solving skills. Keep practicing, keep exploring, and keep having fun with math! You guys did awesome. Until next time!