Find the Equation of a Circle with Given Center and Radius

When it comes to mastering the nuances of collegiate-level algebra, one of the classic shapes that you might find yourself grappling with is the circle. Understanding how to form the equation of a circle based on its center and radius is not just foundational; it's often a crucial part of any College Algebra curriculum. So, let’s break this down together – and don’t worry, you’ll be tossing around these terms and principles with ease in no time!

You’re presented with a problem that asks for the equation of a circle with a center located at the point (2, -1) and a radius of 3. Sounds straightforward, right? Here's the thing: to find the equation of a circle, you’ll want to leverage the standard formula, which is expressed as ((x - h)^2 + (y - k)^2 = r^2). Now, hold on. The 'h' and 'k' in this formula represent the x and y coordinates of the circle’s center, while 'r' signifies the radius.

So, let's fill in the blanks! Given that your center is at (2, -1), that means (h = 2) and (k = -1). Your radius is 3, so you’d set (r = 3). Easy so far, right? Now, let’s plug these values into the equation template. It morphs into:

[ (x - 2)^2 + (y + 1)^2 = 3^2 ]

Wait, hold up. Are you paying attention to the signs here? Since the y-coordinate of the center is -1, the equation uses ((y + 1)) instead of ((y - 1)). And squaring the radius gives you 9. So, our final masterpiece is:

[ (x - 2)^2 + (y + 1)^2 = 9 ]

Now that’s clear, but let's clarify something—after running through a set of multiple-choice options, it appears that the option D aligns perfectly with your findings. It states:

D. ((x - 2)^2 + (y - 1)^2 = 9)

But, there’s a glitch here. If we revisit the other options, it’s apparent that the center in option D is incorrectly represented as ((y - 1)) rather than the correct ((y + 1)). Therefore, while you might be tempted to check option D as correct since the radius is accurate, the real answer lies in the correct representation of both the center and radius.

The correct format, which stays true to our center and radius, should be:

(x - 2)^2 + (y + 1)^2 = 9.

If that seems convoluted, let’s untangle it one last time. The heart of the problem is really about making sure every piece fits snugly together. Remember, the true test lies in understanding how these geometric principles interlace with the equation formats. Got it? Good!

So whether you're gearing up for studying or just brushing up on these essential algebraic skills, remember that the foundation of algebra doesn’t just apply to circles. Connecting these concepts seamlessly is what will set you apart when facing any exam. Plus, once you nail down the equation of a circle, think of all the other geometric wonders waiting at your fingertips. Keep your spirits high and keep practicing!