Equation Of A Circle With Center (1,-3) And Radius 16

by Chloe Fitzgerald 54 views

Hey guys! Today, we're diving into the fascinating world of circles and their equations. Specifically, we're going to figure out which equation perfectly represents a circle with its center snugly placed at the coordinates (1, -3) and stretching out with a radius of 16 units. This is a classic problem in geometry, and understanding how to solve it is super useful for all sorts of math challenges. So, let's get started and break down the mystery behind the equation of a circle!

The General Equation of a Circle

Before we jump into solving the problem, let's quickly recap the general equation of a circle. This is our trusty tool that will guide us to the correct answer. The general form looks like this:

(x - h)² + (y - k)² = r²

Where:

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

Think of this equation as a blueprint for circles. It tells us exactly how to describe a circle if we know its center and radius. The (x - h)² and (y - k)² parts deal with the position of the circle in the coordinate plane, while the r² term dictates its size. So, if we know these three values (h, k, and r), we can write the equation of any circle!

Now, let's dig deeper into why this equation works. It all comes down to the Pythagorean theorem, a fundamental concept in geometry that relates the sides of a right triangle. Imagine drawing a right triangle inside our circle, where the hypotenuse (the longest side) is the radius of the circle. The legs of this triangle are the horizontal and vertical distances from any point on the circle to the center. The Pythagorean theorem tells us that the square of the hypotenuse (r²) is equal to the sum of the squares of the legs [(x - h)² and (y - k)²]. This is precisely what the equation of a circle expresses!

Understanding this connection to the Pythagorean theorem not only helps us remember the equation but also gives us a deeper appreciation for the elegance and interconnectedness of mathematical concepts. It's like seeing the hidden gears that make the clockwork of geometry tick!

So, with our understanding of the general equation solid, we're now fully equipped to tackle the problem at hand. We know the center of our circle is (1, -3) and the radius is 16 units. All that's left is to plug these values into the general equation and see which of the answer choices matches our result. Let's get to it!

Plugging in the Values

Okay, guys, let's get our hands dirty and actually plug in the values we know into the general equation of a circle. Remember, our general equation is:

(x - h)² + (y - k)² = r²

And we have:

  • Center: (h, k) = (1, -3)
  • Radius: r = 16

Now, it's just a matter of substituting these values into the equation. Replacing 'h' with 1, 'k' with -3, and 'r' with 16, we get:

(x - 1)² + (y - (-3))² = 16²

Notice how we've carefully replaced each variable with its corresponding value. It's super important to pay attention to the signs here, especially when dealing with negative numbers. Now, let's simplify this equation a bit.

The double negative in the (y - (-3)) term can be simplified to a positive, giving us:

(x - 1)² + (y + 3)² = 16²

And finally, we need to calculate 16 squared (16 * 16), which is 256. So, our equation becomes:

(x - 1)² + (y + 3)² = 256

This is the equation that represents a circle with a center at (1, -3) and a radius of 16 units. We've successfully translated the geometric information (center and radius) into an algebraic equation. This is a core skill in coordinate geometry, and mastering it opens doors to solving a wide range of problems.

Now, the next step is to compare this equation with the answer choices provided in the original problem. We're looking for the option that exactly matches our derived equation. This is where our careful work in plugging in the values and simplifying the equation pays off. We're not just guessing; we've logically deduced the correct equation. So, let's move on and find the matching answer choice!

Matching with the Answer Choices

Alright, we've done the heavy lifting and derived the equation that represents our circle: (x - 1)² + (y + 3)² = 256. Now comes the satisfying part – matching this equation with the answer choices given in the problem.

Let's quickly recap the answer choices:

A. (x + 1)² + (y + 3)² = 16 B. (x + 1)² + (y - 3)² = 256 C. (x - 1)² + (y + 3)² = 256 D. (x - 1)² + (y - 3)² = 4

Scanning through the options, it's clear that Option C: (x - 1)² + (y + 3)² = 256 perfectly matches the equation we derived. Victory is ours!

Let's quickly analyze why the other options are incorrect:

  • Option A has (x + 1)² instead of (x - 1)², which would indicate a center with an x-coordinate of -1, not 1. Also, the right side is 16, which is the radius squared, not the radius squared (256).
  • Option B has (x + 1)² again, indicating the wrong x-coordinate for the center, and (y - 3)², which indicates the wrong y-coordinate (-3 instead of +3).
  • Option D has the correct (x - 1)² part, but (y - 3)² indicates the wrong y-coordinate for the center, and the right side is 4, which is far too small for a circle with a radius of 16.

So, by carefully comparing our derived equation with the answer choices, we've confidently identified Option C as the correct answer. This highlights the importance of not just finding an answer but also understanding why the other options are wrong. This deeper understanding solidifies our knowledge and helps us avoid common mistakes.

Conclusion: Mastering the Circle Equation

Awesome! We've successfully navigated the world of circles and their equations. We started with a problem asking us to identify the equation of a circle with a specific center and radius, and we systematically broke it down step by step. We revisited the general equation of a circle, understood its connection to the Pythagorean theorem, plugged in the given values, simplified the equation, and finally, matched it with the correct answer choice.

This exercise demonstrates the power of understanding the fundamental concepts in mathematics. By grasping the general equation of a circle and knowing how to apply it, we can confidently solve a wide variety of problems. Remember, mathematics isn't just about memorizing formulas; it's about understanding the relationships between different concepts and applying them logically.

So, what are the key takeaways from this adventure?

  • The general equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
  • This equation is rooted in the Pythagorean theorem, connecting geometry and algebra.
  • Carefully plug in the values, paying attention to signs.
  • Simplify the equation to make it easier to compare with answer choices.
  • Understand why the incorrect options are wrong to solidify your understanding.

With these skills in your mathematical toolkit, you'll be well-equipped to tackle any circle-related challenges that come your way. Keep practicing, keep exploring, and most importantly, keep enjoying the beauty and logic of mathematics! You guys are doing great!