Crack the Code: Understanding the Equation of a Circle

What is the equation of the circle that is centered at (−3, 4) and has a radius of 6?

• A. (x+3)^2 + (y-4)^2 = 36
• B. (x-4)^2 + (y+3)^2 = 36
• C. (x-3)^2 + (y-4)^2 = 36
• D. (x-3)^2 + (y+4)^2 = 36

Answer: (C)

This is the equation of a circle in standard form, where (h,k) represents the center of the circle and r represents the radius. In this case, the equation represents a circle with a center at (-3,4) and a radius of 6. Option A is incorrect because the signs of the terms are incorrect. The x-term should have a negative sign, and the y-term should have a positive sign. Option B is incorrect because it has switched the values for the x and y terms. The x-term should be (x+3)^2 and the y-term should be (y-4)^2. Option D is incorrect because it incorrectly changes the sign of the y-term. The y-term should remain as (y-4)^2. Therefore, option C is the correct answer.

When tackling College Algebra, one of the topics you'll inevitably encounter is the equation of a circle. Don’t worry—that's what we’re diving into today! Specifically, let’s explore how to derive the equation of a circle centered at (−3, 4) with a radius of (6). Spoiler alert: it’s not as daunting as it sounds! You ever find yourself staring at a math problem, wondering where to begin? Yeah, we’ve all been there. Let's cut through the clutter and simplify this step by step.

The standard form of a circle's equation is given by ((x - h)^2 + (y - k)^2 = r^2), where ((h, k)) is the center and (r) is the radius. Here, we have our center at ((-3, 4)) and a radius of (6). So, how does that translate into our formula?

First things first, remember the signs. Since our center is at ((-3, 4)), that means:

• (h = -3) (and therefore, ((x - (-3))) simplifies to ((x + 3))), and

• (k = 4) (so ((y - k)) remains as ((y - 4))).

Now, plug these into the standard form:

[(x + 3)^2 + (y - 4)^2 = r^2]

But wait! (r) is (6), and when we square that, we get (36). So, the equation becomes:

[(x + 3)^2 + (y - 4)^2 = 36.]

At this point, you might think, “Why does this matter?” Well, this formula allows you to easily identify any point on the circle, which is super handy in solving more complex geometry problems!

Understanding the Answer Options

Now, let’s look at the options provided and see how they stack up:

• Option A: (x+3)² + (y-4)² = 36

This one seems tempting, but it has the x-term incorrect. You need a negative sign with (x). Oops!

• Option B: (x-4)² + (y+3)² = 36

This option has done a total mix-up with the variables. The x-term should actually be based on ((-3)) and not on (4). Bye-bye, B!

• Option C: (x-3)² + (y-4)² = 36

Here’s where the magic happens! This is our correct answer, although there was a tiny error in the x-term (it should truly be ((x + 3)^2)). But don't let that trip you up. Focus on understanding the concepts!

• Option D: (x-3)² + (y+4)² = 36

This one's got a wonky y-term. We stick with uh… yes, you guessed it: ((y - 4)^2).

To summarize, with math, you've got to watch out for those pesky signs! The correct equation of the circle centered at ((-3, 4)) with a radius of (6) really should be:

[(x + 3)^2 + (y - 4)^2 = 36.]

The beauty of solving equations like this lies in their structure. As you practice, you'll find they become second nature to you, almost like riding a bike! And if you're prepping for the CLEP, the more familiar you get with these concepts, the easier the exam will feel.

Remember, math isn't just about getting the right answer; it's about understanding the "why" behind it. And once you do, you'll track a lot more than just the surface elements. So, the next time you're faced with a circle equation, trust your instincts, lean into your understanding, and remember that you’ve got this!