Understanding the Equation of a Circle: A Clear Guide

What is the equation of a circle if the center is (4,2) and the radius is 3?

• A. (x - 4)2 + (y-2)2 = 9
• B. (x - 4)2 + (y+2)2 = 9
• C. (x + 4)2 + (y-2)2 = 9
• D. (x + 4)2 + (y+2)2 = 9

Answer: (A)

For a circle with center (h,k) and radius r, the equation is (x-h)^2 + (y-k)^2 = r^2. Since the center of this circle is (4,2) and the radius is 3, the equation becomes (x-4)^2 + (y-2)^2 = 9. Option A is the only one that follows this equation, making it the correct answer. Options B, C, and D all have incorrect signs in front of the y term, which would result in a different circle with a different center point.

Have you ever wondered how the equation of a circle comes together? It's like a dance of numbers and symbols—each piece matters. Let's break it down in a way that clicks!

When you think of a circle, you might picture perfect symmetry and a smooth ride around that infinite loop. But behind that beautiful shape lies a simple, yet powerful, equation: ((x - h)^2 + (y - k)^2 = r^2). Here, ((h, k)) is the center of the circle, and (r) represents the radius. If you take a look at the center coordinates ((4, 2)) and a radius of (3), it might seem a bit intimidating at first, but with a little confidence, you can easily find the equation that describes this circle.

Now, plug in what you know: the center is at ((h, k)=(4, 2)) and (r=3). So, if we rearrange the equation using these values, it’s smooth sailing from here! You’d find:

[

(x - 4)^2 + (y - 2)^2 = 3^2,

]

which simplifies to

[

(x - 4)^2 + (y - 2)^2 = 9.

]

Look—did you catch that? The answer is Option A: ((x - 4)^2 + (y - 2)^2 = 9). This is the formula you were searching for!

Let’s pause and consider the other options. They each have their own twist, all trying to claim they could fit the circle we're looking for. But don’t be fooled!

• Option B ((x - 4)^2 + (y + 2)^2 = 9) suggests a center at ((4, -2))—definitely not right for our situation.

• Option C ((x + 4)^2 + (y - 2)^2 = 9) throws the center off to ((-4, 2))—another twist that doesn’t belong in our equation.

• Finally, Option D ((x + 4)^2 + (y + 2)^2 = 9) uses a center at ((-4, -2))—which is a whole different ballgame.

It's fascinating how just a change in signs can send the entire equation spiraling in a new direction!

You see, writing equations for circles isn't just about memorizing formulas—it's about tapping into the rhythm behind the math. Once you grasp why we use ((x - h)) and ((y - k)), you'll feel much more at ease tackling circles in any algebra setting. It’s a bit like recognizing the notes in a song; once you catch on, it all starts to flow.

If you’re prepping for that College Algebra CLEP Exam, just remember: it all comes down to recognizing patterns, understanding the relationships in your equations, and practicing until it's second nature. Who knew math could feel so much like a game? Keep those gears turning, and good luck on your tests!