Mastering Circle Equations for the College Algebra CLEP Exam

Which of the following is an equation of the circle with center (3,2) and a radius of 5?

• A. x^2 + y^2 = 7
• B. (x - 3)^2 + (y - 2)^2 = 7
• C. (x - 3)^2 + (y - 2)^2 = 25
• D. x^2 + y^2 = 25

Answer: (C)

Option A is incorrect because the radius is not squared and does not match the given circle's radius of 5. Option B is also incorrect because the equation is missing the radius and it is not centered at (3,2). Option D is incorrect because the equation is not centered at (3,2). The correct equation must have the center coordinates, followed by the radius squared. Therefore, option C, (x-3)^2 + (y-2)^2 = 25, is the correct equation. This allows for a circle with the center at (3,2) and a radius of 5, because when the coordinates are substituted in, the equation becomes (3-3)^2 + (2-2)^2 = 0 + 0 = 25, which satisfies the equation of a circle, (x-h)^2 + (y-k

Understanding the equations of circles can feel a bit like deciphering code. Trust me, it’s easier than it looks! Your College Algebra CLEP Prep requires you to identify the equation of a circle based on its center and radius. If you're studying for this exam, get ready to channel your inner math whiz as we break down a sample problem that you just might see on test day.

What’s the Deal with Circle Equations?

You know what? The equation of a circle is more than just numbers and letters thrown onto a page. It’s a beautiful representation of geometry in a mathematical form. The general equation can be expressed as ((x - h)^2 + (y - k)^2 = r^2), where ( (h, k) ) represents the center of the circle, and ( r ) is the radius. Let’s break this down: the center is like the heart of the circle, and the radius is the heartbeat, stretching outward.

Let’s Solve a Circle Equation Together

Alright, let’s roll up our sleeves and work on a typical question together—one that you might run into while preparing for your exam:

Question: Which of these equations represents a circle with center ( (3, 2) ) and a radius of ( 5 )?

A. ( x^2 + y^2 = 7 )

B. ( (x - 3)^2 + (y - 2)^2 = 7 )

C. ( (x - 3)^2 + (y - 2)^2 = 25 )

D. ( x^2 + y^2 = 25 )

Identifying the Correct Answer

Let’s dissect these options together. First off, the center of the circle is given by ( (3, 2) ) — this means our equation should reflect that. The radius is ( 5 ), so we need to square that to get ( 25 ) (because ( r^2 ) is crucial here!).

Now, let’s consider each option:

• A. ( x^2 + y^2 = 7 ) — Clearly off base. The radius isn't even squared, plus it doesn't center around ( (3, 2) ) at all.

• B. ( (x - 3)^2 + (y - 2)^2 = 7 ) — A bit closer, but it’s still lacking. It misses the correct radius squared and doesn't center at our point.

• C. ( (x - 3)^2 + (y - 2)^2 = 25 ) — Ding, ding, ding! This checks all the boxes. It centers at ( (3, 2) ) and uses ( 25 ) (since ( 5^2 = 25 )). This is the right equation for our circle.

• D. ( x^2 + y^2 = 25 ) — While it does give a radius that looks correct, once again, it fails to center around ( (3, 2) ).

Why Option C is the Gold Star Answer

So why is option C the golden child of equations here? Because it’s built on the correct understanding of the center-radius relationship. When you plug in ( (3, 2) ) into the correct equation, it confirms:

((3-3)^2 + (2-2)^2 = 0 + 0 = 25)

This satisfies our original circle equation. It’s like double-checking your homework before turning it in—you’re ensuring that everything aligns perfectly.

Final Thoughts Before You Head Out

Now, take a moment to reflect: understanding these equations lays the groundwork not just for this problem but for many similar concepts in algebra. It’s not only about getting the correct answer (though that’s important too), but about grasping the theory behind it all!

Feeling that weight lift off your shoulders? Yeah, mastering these concepts will boost your confidence in your College Algebra journey. Trust me, when you walk into that exam room, you’ll be ready to shine!

So grab a cup of coffee, hit the books, and keep practicing. You’ll ace this!